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With the ytt4&k&s Compliments. 



MATTER AND MOTION 



MATTER AND MOTION/ 



BY THE LATE 

J. CLERK JV^^XWELL 

M.A., LL.D.EDIN., F.R.SS.L. & E. 

HONORARY FELLOW OF TRINITY COLLEGE, AND PROFESSOR OF 
EXPERIMENTAL PHYSICS IN THE UNIVERSITY OF CAMBRIDGE 



REPRINTED : WITH NOTES AND APPENDICES BY 
SIR JOSEPH LARMOR, F.R.S., M.P. 

FELLOW OF ST JOHN'S COLLEGE, AND 
LUCASIAN PROFESSOR OF MATHEMATICS 



LONDON : 

SOCIETY FOR PROMOTING 

CHRISTIAN KNOWLEDGE 

NEW YORK : THE MACMILLAN CO. 

1920 



PREFACE (1877) 

PHYSICAL SCIENCE, which up to the end of the eighteenth 
century had been fully occupied in forming a conception 
of natural phenomena as the result of forces acting 
between one body and another, has now fairly entered 
on the next stage of progress that in which the energy 
of a material system is conceived as determined by the 
configuration and motion of that system, and in which 
the ideas of configuration, motion, and force are 
generalised to the utmost extent warranted by their 
physical definitions. 

To become acquainted with these fundamental ideas, 
to examine them under all their aspects, and habitually 
to guide the current of thought along the channels of 
strict dynamical reasoning, must be the foundation of 
the training of the student of Physical Science. 

The following statement of the fundamental doctrines 
of Matter and Motion is therefore to be regarded as 
an introduction to the study of Physical Science in 
general. 



NOTE 

IN this reprint of Prof. Clerk Maxwell's classical 
tractate on the principles of dynamics, the changes have 
been confined strictly to typographical and a few verbal 
improvements. After trial, the conclusion has been 
reached that any additions to the text would alter the 
flavour of the work, which would then no longer be 
characteristic of its author. Accordingly only brief 
footnotes have been introduced: and the few original 
footnotes have been distinguished from them by 
Arabic numeral references instead of asterisks and other 
marks. A new index has been prepared. 

A general exposition of this kind cannot be expected, 
and doubtless was not intended, to come into use as a 
working textbook : for that purpose methods of syste- 
matic calculation must be prominent. But as a reasoned 
conspectus of the Newtonian dynamics, generalizing 
gradually from simple particles of matter to physical 
systems which are beyond complete analysis, drawn 
up by one of the masters of the science, with many 
interesting side-lights, it must retain its power of sug- 
gestion even though parts of the vector exposition may 
now seem somewhat abstract. The few critical footnotes 
and references to Appendices that have been added may 
help to promote this feature of suggestion and stimulus. 

The treatment of the fundamental principles of 
dynamics has however been enlarged on the author's 
own lines by the inclusion of the Chapter "On the 
Equations of Motion of a Connected System" from 
vol. ii of Electricity and Magnetism. For permission to 
make use of this chapter the thanks of the publishers 
are due to the Clarendon Press of the University of 
Oxford. 



viii NOTE 

With the same end in view two Appendices have 
been added by the editor. One of them treats the 
Principle of Relativity of motion, which has recently 
become very prominent in wider physical connexions, 
on rather different lines from those in the text. The other 
aims at development of the wider aspects of the Prin- 
ciple of Least Action, which has been asserting its 
position more and more as the essential principle of con- 
nexion between the various domains of Theoretical 
Physics. 

These additions are of course much more advanced 
than the rest of the book : but they will serve to complete 
it by presenting the analytical side of dynamical science, 
on which it justly aspires to be the definite foundation 
for all Natural Philosophy. 

The editor desires to express his acknowledgment 
to the Cambridge University Press, and especially to 
Mr J. B. Peace, for assistance and attention. 

J.L. 



BIOGRAPHICAL NOTE 

JAMES CLERK MAXWELL was born in Edinburgh in 1831, 
the only son of John Clerk Maxwell, of Glenlair, near 
Dalbeattie, a family property in south-west Scotland to 
which the son succeeded. After an early education at 
home, and at the University of Edinburgh, he pro- 
ceeded to Cambridge in 1850, first to Peterhouse, 
migrating afterwards to Trinity College. In the 
Mathematical Tripos of 1854, the Senior Wrangler was 
E. J. Routh, afterwards a mathematical teacher and 
investigator of the highest distinction, and Clerk Max- 
well was second: they were placed as equal soon after 
in the Smith's Prize Examination. 

He was professor of Natural Philosophy at Aberdeen 
from 1856 to 1860, in King's College, London from 
1860 to 1865, and then retired to Glenlair for six years, 
during which the teeming ideas of his mind doubtless 
matured and fell into more systematic forms. He was 
persuaded to return into residence at Cambridge in 
1871, to undertake the task of organizing the new 
Cavendish Laboratory. But after a time his health 
broke, and he died in 1879 at the age of 48 years. 

His scientific reputation during his lifetime was 
upheld mainly by British mathematical physicists, 
especially by the Cambridge school. But from the time 
that Helmholtz took up the study of his theory of 
electric action and light in 1870, and discussed it in 
numerous powerful memoirs, the attention given abroad 
to his work gradually increased, until as in England it 
became the dominating force in physical science. 

Nowadays by universal consent his ideas, as the 
mathematical interpreter and continuator of Faraday, 
rank as the greatest advance in our understanding of 
the laws of the physical universe that has appeared 



x . BIOGRAPHICAL NOTE 

since the time of Newton. As with Faraday, his pro- 
found investigations into nature were concomitant with 
deep religious reverence for nature's cause. See the 
Life by L. Campbell and W. Garnett (Macmillan, 1882). 
The treatise on Electricity and Magnetism and the 
Theory of Heat contain an important part of his work. 
His Scientific Papers were republished by the Cam- 
bridge University Press in two large memorial volumes. 
There are many important letters from him in the 
Memoir and Scientific Correspondence of Sir George 
Stokes, Cambridge, 1904. 

The characteristic portrait here reproduced, perhaps 
for the first time, is from a carte de visile photograph 
taken probably during his London period. 

J. L. 



CONTENTS 



CHAPTER I 
INTRODUCTION 

ART. PAGE 

1 Nature of Physical Science i 

2 Definition of a Material System 2 

3 Definition of Internal and External .... 2 

4 Definition of Configuration 2 

5 Diagrams 3 

6 A Material Particle 3 

7 Relative Position of two Material Particles ... 4 

8 Vectors 4 

9 System of Three Particles 5 

10 Addition of Vectors 5 

1 1 Subtraction of one Vector from another ... 6 

12 Origin of Vectors 6 

13 Relative Position of Two Systems 7 

14 Three Data for the Comparison of Two Systems . . 7 

15 On the Idea of Space 9 

16 Error of Descartes 9 

17 On the Idea of Time n 

18 Absolute Space 12 

19 Statement of the General Maxim of Physical Science . 13 

CHAPTER II 
ON MOTION 

20 Definition of Displacement . . . , . . . 15 

21 Diagram of Displacement 15 

22 Relative Displacement 16 

23 Uniform Displacement 17 

24 On Motion 18 

25 On the Continuity of Motion ... . 18 

26 On Constant Velocity .... . 19 

27 On the Measurement of Velocity when Variable . 19 

28 Diagram of Velocities . . . ... 20 

29 Properties of the Diagram of Velocities . 21 

30 Meaning of the Phrase "At Rest" . . . 22 

31 On Change of Velocity 22 

32 On Acceleration . 23 

33 On the Rate of Acceleration ... . . 24 

34 Diagram of Accelerations . . . ... . 25 

35 Acceleration a Relative Term . . . * . . 25 



xii CONTENTS 



CHAPTER III 
ON FORCE 

ART. PAGE 

36 Kinematics and Kinetics 26 

37 Mutual Action between Two Bodies Stress . . 26 

38 External Force 26 

39 Different Aspects of the same Phenomenon . . . 27 

40 Newton's Laws of Motion ...... 27 

41 The First Law of Motion 28 

42 On the Equilibrium of Forces 30 

43 Definition of Equal Times 31. 

44 The Second Law of Motion 32 

45 Definition of Equal Masses and of Equal Forces . 32 

46 Measurement of Mass 33 

47 Numerical Measurement of Force 35 

48 Simultaneous Action of Forces on a Body ... 36 

49 On Impulse 37 

50 Relation between Force and Mass 38 

51 On Momentum 38 

52 Statement of the Second Law of Motion in Terms of 

Impulse and Momentum . . . . . . 39 

53 Addition of Forces . . . . . . . 39 

54 The Third Law of Motion 40 

55 Action and Reaction are the Partial Aspects of a Stress 40 

56 Attraction and Repulsion . . . ' . . . 41 

57 The Third Law True of Action at a Distance . . 42 

58 Newton's Proof not Experimental ..... 42 



CHAPTER IV 

ON THE PROPERTIES OF THE CENTRE OF MASS 
OF A MATERIAL SYSTEM 

59 Definition of a Mass- Vector ...... 44 

60 Centre of Mass of Two Particles . . . . 44 

6 1 Centre of 'Mass of a System 45 

62 Momentum represented as the Rate of Change of a 

Mass- Vector 45 

63 Effect of External Forces on the Motion of the Centre 

of Mass 46 

64 The Motion of the Centre of Mass of a System is not 

affected by the Mutual Action of the Parts of the 

System 47 

65 First and Second Laws of Motion .... 48 

66 Method of treating Systems of Molecules . .- . 48 



CONTENTS xiii 

ART. PAGE 

67 By the Introduction of the Idea of Mass we pass from 

Point- Vectors, Point Displacements, Velocities, 
. Total Accelerations, and Rates of Acceleration, to 
Mass- Vectors, Mass Displacements, Momenta, Im- 
pulses, and Moving Forces 49 

68 Definition of a Mass- Area 50 

69 Angular Momentum 51 

70 Moment of a Force about a Point 51 

71 Conservation of Angular Momentum .... 52 



CHAPTER V 
ON WORK AND ENERGY 

72 Definitions 54 

73 Principle of Conservation of Energy .... 54 

74 General Statement of the Principle of the Conservation 

of Energy 55 

75 Measurement of Work 56 

76 Potential Energy 58 

77 Kinetic Energy 58 

78 Oblique Forces 60 

79 Kinetic Energy of Two Particles referred to their 

Centre of Ma'ss 61 

80 Kinetic Energy of a Material System referred to its 

Centre of Mass 62 

81 Available Kinetic Energy 63 

82 Potential Energy 65 

83 Elasticity 65 

84 Action at a Distance 66 

85 Theory of Potential Energy more complicated than 

that of Kinetic Energy 67 

86 Application of the Method of Energy to the Calculation 

of Forces 68 

87 Specification of the [Mode of Action] of Forces . . 69 

88 Application to a System in Motion .... 70 

89 Application of the Method of Energy to the Investigation 

of Real Bodies j(B 

90 Variables on which the Energy depends . . . 71 

91 Energy in Terms of the Variables 72 

92 Theory of Heat 72 

93 Heat a Form of Energy . 73 

94 Energy Measured as Heat . . . ... 73 

95 Scientific Work to be done 74 

96 History of the Doctrine of Energy . . . 75 

97 On the Different Forms of Energy . . , . . 76 



CONTENTS 



CHAPTER VI 
RECAPITULATION 

ART. PAGE 

98 Retrospect of Abstract Dynamics . . . . . 79 

99 Kinematics 79 

100 Force .79 

101 Stress 80 

102 Relativity of Dynamical Knowledge .... 80 

103 Relativity of Force 81 

104 Rotation . . 83 

105 Newton's Determination of the Absolute Velocity of 

Rotation . . . . .'.:. . . . 84 

106 Foucault's Pendulum 86 

107 Matter and Energy 89 

108 Test of a Material Substance 89 

109 Energy not capable of Identification .... 90 
no Absolute Value of the Energy of a Body unknown . 90 
in Latent Energy 91 

112 A Complete Discussion of Energy would include the 

whole of Physical Science 91 

CHAPTER VII 
THE PENDULUM AND GRAVITY 

113 On Uniform Motion in a Circle 92 

114 Centrifugal Force 93 

115 Periodic Time 93 

116 On Simple Harmonic Vibrations . . ... . 94 

117 On the Force acting on the Vibrating Body . . 94 

118 Isochronous Vibrations 95 

119 Potential Energy of the Vibrating Body . . , . 96 

1 20 The Simple Pendulum . . . ..... 96 

121 A Rigid Pendulum 98 

122 Inversion of the Pendulum 100 

123 Illustration of Kater's Pendulum . . ... . 100 

124 Determination of the Intensity of Gravity . . . 101 

125 Method of Observation 102 

126 Estimation of Error 103 

CHAPTER VIII 
UNIVERSAL GRAVITATION 

127 Newton's Method . . 105 

128 Kepler's Laws , . 105 

129 Angular Velocity .106 

130 Motion about the Centre of Mass . . . ; .. , . 106 



CONTENTS xv 

ART. PAGE 

131 The Orbit . 107 

. 107 
. 108 
. 109 
no 
. Ill 

112 



132 The Hodograph 

133 Kepler's Second Law 

134 Force on a Planet 

135 Interpretation of Kepler's Third Law . 

136 Law of Gravitation 

137 Amended Form of Kepler's Third Law 

138 Potential Energy due to Gravitation 

139 Kinetic Energy of the System 

140 Potential Energy of the System . . . .114 

141 The Moon is a Heavy Body 115 

142 Cavendish's Experiment 116 

143 The Torsion Balance 117 

144 Method of the Experiment 118 

145 Universal Gravitation 119 

146 Cause of Gravitation 120 

147 Application of Newton's Method of Investigation . 121 

148 Methods of Molecular Investigations .... 122 

149 Importance of General and Elementary Properties . 122 

[CHAPTER IX] 

ON THE EQUATIONS OF MOTION OF A CON- 
NECTED SYSTEM 123 



APPENDIX I 
THE RELATIVITY OF THE FORCES OF NATURE 137 

APPENDIX II 
THE PRINCIPLE OF LEAST ACTION . . .145 

INDEX I62 



Portrait of Prof. CLERK MAXWELL . . Frontispiece 



MATTER AND MOTION 
CHAPTER I 

INTRODUCTION 

i . NATURE OF PHYSICAL SCIENCE 
PHYSICAL SCIENCE is that department of knowledge 
which relates to the order of nature, or, in other words, 
to the regular succession of events. 

The name of physical science, however, is often 
applied in a more or less restricted manner to those 
branches of science in which the phenomena considered 
are of the simplest and most abstract kind, excluding 
the consideration of the more complex phenomena, such 
as those observed in living beings. 

The simplest case of all is that in which an event 
or phenomenon can be described as a change in the 
arrangement of certain bodies. Thus the motion of the 
moon may be described by stating the changes in her 
position relative to the earth in the order in which they 
follow one another. 

In other cases we may know that some change of 
arrangement has taken place, but we may not be able 
to ascertain what that change is. 

Thus when water freezes we know that the molecules 
or smallest parts of the substance must be arranged 
differently in ice and in water. We also know that this 
arrangement in ice must have a certain kind of sym- 
metry, because the ice is in the form of symmetrical 
crystals, but we have as yet no precise knowledge of 
the actual arrangement of the molecules in ice. But 
whenever we can completely describe the change of 



2 INTRODUCTION [CH. 

arrangement we have a knowledge, perfect so far as it 
extends, of what has taken place, though we may still 
have to learn the necessary conditions under which 
a similar event will always take place. 

Hence the first part of physical science relates to the 
relative position and motion of bodies. 

2. DEFINITION OF A MATERIAL SYSTEM 
In all scientific procedure we begin by marking out a 
certain region or subject as the field of our investiga- 
tions. To this we must confine our attention, leaving 
the rest of the universe out of account till we have 
completed the investigation in which we are engaged. 
In physical science, therefore, the first step is to define 
clearly the material system which we make the subject 
of our statements. This system may be of any degree 
of complexity. It may be a single material particle, a 
body of finite size, or any number of such bodies, and 
it may even be extended so as to include the whole 
material universe. 

3. DEFINITION OF INTERNAL AND EXTERNAL 

All relations or actions between one part of this sys- 
tem and another are called Internal relations or actions. 

Those between the whole or any part of the system 
and bodies not included in the system are called Exter- 
nal relations or actions. These we study only so far as 
they affect the system itself, leaving their effect on 
external bodies out of consideration. Relations and 
actions between bodies not included in the system are 
to be left out of consideration. We cannot investigate 
them except by making our system include these other 
bodies. 

4. DEFINITION OF CONFIGURATION 

When a material system is considered with respect 
to the relative position of its parts, the assemblage of 
relative positions is called the Configuration of the 
system. 



i] CONFIGURATION 3 

A knowledge of the configuration of the system at a 
given instant implies a knowledge of the position of 
every point of the system with respect to every other 
point at that instant. 

5. DIAGRAMS 

The configuration of material systems may be repre- 
sented in models, plans, or diagrams. The model or 
diagram is supposed to resemble the material system 
only in form, not necessarily in any other respect. 

A plan or a map represents on paper in two dimen- 
sions what may really be in three dimensions, and can 
only be completely represented by a model. We shall 
use the term Diagram to signify any geometrical figure, 
whether plane or not, by means of which we study the 
properties of a material system. Thus, when we speak 
of the configuration of a system, the image which we 
form in our minds is that of a diagram, which completely 
represents the configuration, but which has none of the 
other properties of the material system. Besides dia- 
grams of configuration we may have diagrams of velocity, 
of stress, etc., which do not represent the form of the 
system, but by means of which its relative velocities or 
its internal forces may be studied. 

6. A MATERIAL PARTICLE 

A body so small that, for the purposes of our investi- 
gation, the distances between its different parts may be 
neglected, is called a material particle. 

Thus in certain astronomical investigations the planets, 
and even the sun, may be regarded each as a material 
particle, because the difference of the actions of different 
parts of these bodies does not come under our notice. 
But we cannot treat them as material particles when we 
investigate their rotation. Even an atom, when we 
consider it as capable of rotation, must be regarded as 
consisting of many material particles. 

The diagram of a material particle is of course a 
mathematical point, which has no configuration. 



4 INTRODUCTION [CH. 

7. RELATIVE POSITION OF TWO MATERIAL PARTICLES 

The diagram of two material particles consists of two 
points, as, for instance, A and B. 

The position of B relative to A is indicated by the 
direction and length of the straight line AB drawn 
from A to B. If you start from A and travel in the 
direction indicated by the line AB and for a distance 
equal to the length of that line, you will get to B. 
This direction and distance may be indicated equally 
well by any other line, such as ab, which is parallel 
and equal to AB. The position of A with respect to 
B is indicated by the direction and length of the line 
BA, drawn from B to A, or the line ba, equal and 
parallel to BA. 

It is evident that BA = AB. 

In naming a line by the letters at its extremities, 
the order of the letters is always that in which the line 
is to be drawn. 

8. VECTORS 

The expression AB, in geometry, is merely the 
name of a line. Here it indicates the operation by 
which the line is drawn, that of carrying a tracing 
point in a certain directionj:or a certain distance. As 
indicating an operation, AB is called a Vector, and 
the operation is completely defined by the direction 
and distance of the transference. The starting point, 
which is called the Origin of the vector, may be any- 
where. 

To define a finite straight line we must state its 
origin as well as its direction and length. All vectors, 
however, are regarded as equal which are parallel (and 
drawn towards the same parts) and of the same magni- 
tude. 

Any quantity, such, for instance, as a velocity or a 



i] VECTORS 5 

t. - . 4 

force*, which has a definite direction and a definite 
magnitude may be treated as a vector, and may 
be indicated in a diagram by a straight line whose 
direction is parallel to the vector, and whose length 
represents, according to a determinate scale, the mag- 
nitude of the vector. 

9. SYSTEM OF THREE PARTICLES 

Let us next consider a system of three particles. 

Its configuration is represented by a diagram of 
three points, A, B, C. 

The position of B with respect to D. C 

A is indicated by the vector AB, I // 
and that of C with respect to B by A/ I 
the vector EC. A *- 'B 

It is manifest that from these data, Fig. i. 

when A is known, we can find B and 
then C, so that the configuration of the three points is 
completely determined. 

The position of C with respect to A is indicated by 
the vector AC, and by the last remark the value of AC 
must be deducible from those of AB and BC. 

The result of the operation AC is to carry the 
tracing point from A to C. But the result is the same 
if the tracing point is carried first from A to B and 
then from B to C, and this is the sum of the operations 
AB + BC. 

10. ADDITION OF VECTORS 

Hence the rule for the addition of vectors may be 
stated thus: From any point as origin draw the suc- 
cessive vectors in series, so that each vector begins at 
the end of the preceding one. The straight line from 
the origin to the extremity of the series represents the 
vector which is the sum of the vectors. 

* A force is more completely specified as a vector localised in 
its line of action, called by Clifford a rotor; moreover it is only 
when the body on which it acts is treated as rigid that the point 
of application is inessential. 



6 INTRODUCTION [CH. 

The order of addition is indifferent, for if we write 
BC + AB the_pperation indicated may be performed 
by drawing AD parallel and equal to BC, and then 
joining DC, which, by Euclid, I. 33, is parallel and 
equal to AB, so that by these two operations we arrive 
at the point C in whichever order we perform them. 

The same is true for any number of vectors, take 
them in what order we please. 

1 1 . SUBTRACTION OF ONE VECTOR FROM ANOTHER 
To express the position of C with respect to B in 
terms of the positions of B and C with respect to A, 
we observe that we can get from B to C either by 
passing along the straight line BC or by passing from 
B to A and then from A to C. Hence 



= AC + BA since the order of addition is indifferent 
= AC AB since AB is equal and opposite to BA . 

Or the vector BC, which expresses the position of C 
with respect to B, is found by subtracting the vector of 
B from the vector of C, these vectors being drawn to 
B and C respectively from any common origin A. 

12. ORIGIN OF VECTORS 

The positions of any number of particles belonging 
to a material system may be defined by means of the 
vectors drawn to each of these particles from some one 
point. This point is called the origin of the vectors, 
or, more briefly, the Origin. 

This system of vectors determines the configura- 
tion of the whole system; for if we wish to know 
the position of any point B with respect to any other 
point A, it may be found from the vectors OA and OB 
by the equation 

AB=OB-OA. 



i] RELATIVE POSITION 7 

We may choose any point whatever for the origin, 
and there is for the present no reason why we should 
choose one point rather than another. The configura- 
tion of the system that is to say, the position of its 
parts with respect to each other remains the same, 
whatever point be chosen as origin. Many inquiries, 
however, are simplified by a proper selection of the 
origin. 

13. RELATIVE POSITION OF Two SYSTEMS 

If the configurations of two different systems are 
known, each system having its own 
origin, and if we then wish to include \ 
both systems in a larger system, 
having, say, the same origin as the _, 

first of the two systems, we must 
ascertain the position of the origin of 
the second system with respect to that of the first, and 
we must be able to draw lines in the second system 
parallel to those in the first. 

Then by Article 9 the position of a point P of the 
second system, with respect to the first origin, O, is 
represented by the sum of the vector O'P of that point 
with respect to the second origin, O', and the vector OO' 
of the second origin, O', with respect to the first, O. 

14. THREE DATA FOR THE COMPARISON OF 
Two SYSTEMS 

We have an instance of this formation of a large 
system out of two or more smaller systems, when two 
neighbouring nations, having each surveyed and 
mapped its own territory, agree to connect their sur- 
veys so as to include both countries in one system. 
For this purpose three things are necessary. 

i st. A comparison of the origin selected by the one 
country with that selected by the other. 

2nd. A comparison of the directions of reference 
used in the two countries. 



8 INTRODUCTION [CH. 

3rd. A comparison of the standards of length used 
in the two countries. 

1. In civilised countries latitude is always reckoned 
from the equator, but longitude is reckoned from an 
arbitrary point, as Greenwich or Paris. Therefore, 
to make the map of Britain fit that of France, we 
must ascertain the difference of longitude between the 
Observatory of Greenwich and that of Paris. 

2. When a survey has been made without astro- 
nomical instruments, the directions of reference have 
sometimes been those given by the magnetic compass. 
This was, I beb'eve, the case in the original surveys of 
some of the West India islands. The results of this 
survey, though giving correctly the local configuration 
of the island, could not be made to fit properly into a 
general map of the world till the deviation of the 
magnet from the true north at the time of the survey 
was ascertained. 

3. To compare the survey of France with that of 
Britain, the metre, which is the French standard of 
length, must be compared with the yard, which is the 
British standard of length. 

The yard is defined by Act of Parliament 18 and 
19 Viet. c. 72, July 30, 1855, which enacts "that the 
straight line or distance between the centres of the 
transverse lines in the two gold plugs in the bronze 
bar deposited in the office of the Exchequer shall 
be the genuine standard yard at 62 Fahrenheit, 
and if lost, it shall be replaced by means of its copies." 

The metre derives its authority from a law of the 
French Republic in 1795. It is defined to be the 
distance between the ends of a certain rod of platinum 
made by Borda, the rod being at the temperature of 
melting ice. It has been found by the measurements 
of Captain Clarke that the metre is equal to 39-37043 
British inches. 



i] SPACE 9 

15. ON THE IDEA OF SPACE* 

We have now gone through most of the things to be 
attended to with respect to the configuration of a 
material system. There remain, however, a few points 
relating to the metaphysics of the subject, which have a 
very important bearing on physics. 

We have described the method of combining several 
configurations into one system which includes them all. 
In this way we add to the small region which we can 
explore by stretching our limbs the more distant regions 
which we can reach by walking or by being carried. 
To these we add those of which we learn by the reports 
of others, and those inaccessible regions whose positions 
we ascertain only by a process of calculation, till at last 
we recognise that every place has a definite position 
with respect to every other place, whether the one 
place is accessible from the other or not. 

Thus from measurements made on the earth's surface 
we deduce the position of the centre of the earth relative 
to known objects, and we calculate the number of 
cubic miles in the earth's volume quite independently 
of any hypothesis as to what may exist at the centre of 
the earth, or in any other place beneath that thin layer 
of the crust of the earth which alone we can directly 
explore. 

1 6. ERROR OF DESCARTES 

It appears, then, that the distance between one thing 
and another does not depend on any material thing 
between them, as Descartes seems to assert when he 
says (Princip. Phil., II. 18) that if that which is in a 
hollow vessel were taken out of it without anything 

* Following Newton's method of exposition in the Principia, 
a space is assumed and a flux of time is assumed, forming together 
a framework into which the dynamical explanation of phenomena 
is set. It is part of the problem of physical astronomy to test this 
assumption, and to determine this frame with increasing precision. 
Its philosophical basis can be regarded as a different subject, to 
which the recent discussions on relativity as regards space and 
time would be attached. See Appendix I. 



io INTRODUCTION [CH. 

entering to fill its place, the sides of the vessel, having 
nothing between them, would be in contact. 

This assertion is grounded on the dogma of Des- 
cartes, that the extension in length, breadth, and depth 
which constitute space is the sole essential property of 
matter. "The nature of matter," he tells us, "or of 
body considered generally, does not consist in a thing 
being hard, or heavy, or coloured, but only in its 
being extended in length, breadth, and depth " (Princip., 
II. 4). By thus confounding the properties of matter 
with those of space, he arrives at the logical conclusion 
that if the matter within a vessel could be entirely 
removed, the space within the vessel would no longer 
exist. In fact he assumes that all space must be always 
full of matter. 

I have referred to this opinion of Descartes in order 
to show the importance of sound views in elementary 
dynamics. The primary property of matter was in- 
deed distinctly announced by Descartes in what he 
calls the "First Law of Nature" (Princip., II. 37): 
"That every individual thing, so far as in it lies, per- 
severes in the same state, whether of motion or of rest."* 

We shall see when we come to Newton's laws of 
motion that in the words "so far as in it lies," pro- 
perly understood, is to be found the true primary 
definition of matter, and the true measure of its quantity. 
Descartes, however, never attained to a full under- 
standing of his own words (quantum in se esf), and so 
fell back on his original confusion of matter with space 
space being, according to him, the only form of 
substance, and all existing things but affections of space. 
This errorf runs through every part of Descartes' great 
work, and it forms one of the ultimate foundations of 
the system of Spinoza. I shall not attempt to trace 
it down to more modern times, but I would advise 

* Compare the idea of Least Action: Appendix II. 
f Some recent forms of relativity have come back to his ideas. 
Cf. p. 140. 



i] TIME ii 

those who study any system of metaphysics to examine 
carefully that part of it which deals with physical ideas. 
We shall find it more conducive to scientific pro- 
gress to recognise, with Newton, the ideas of time and 
space as distinct, at least in thought, from that of the 
material system whose relations these ideas serve to 
co-ordinate*. 

17. ON THE IDEA OF TIME 

The idea of Time in its most primitive form is pro- 
bably the recognition of an order of sequence in our 
states of consciousness. If my memory were perfect, I 
might be able to refer every event within my own 
experience to its proper place in a chronological series. 
But it would be difficult, if not impossible, for me to 
compare the interval between one pair of events and 
that between another pair to ascertain, for instance, 
whether the time during which I can work without 
feeling tired is greater or less now than when I first 
began to study. By our intercourse with other persons, 
and by our experience of natural processes which go 
on in a uniform or a rhythmical manner, we come 
to recognise the possibility of arranging a system of 
chronology in which all events whatever, whether re- 
lating to ourselves or to others, must find their places. 
Of any two events, say the actual disturbance at the 
star in Corona Borealis, which caused the luminous 
effects examined spectroscopically by Mr Huggins on 
the 1 6th May, 1866, and the mental suggestion which 
first led Professor Adams or M. Leverrier to begin the 
researches which led to the discovery, by Dr Galle, on 
the 23rd September, 1846, of the planet Neptune, the 
first named must have occurred either before or after 
the other, or else at the same time. 

Absolute, true, and mathematical Time is conceived 
by Newton as flowing at a constant rate, unaffected by 
the speed or slowness of the motions of material things. 

* See Appendix I. 



12 INTRODUCTION [CH. 

It is also called Duration. Relative, apparent, and 
common time is duration as estimated by the motion 
of bodies, as by days, months, and years. These 
measures of time may be regarded as provisional, for 
the progress of astronomy has taught us to measure the 
inequality in the lengths of days, months, and years, 
and thereby to reduce the apparent time to a more 
uniform scale, called Mean Solar Time. 

1 8. ABSOLUTE SPACE 

Absolute space is conceived as remaining always 
similar to itself and immovable. The arrangement 
of the parts of space can no more be altered than the 
order of the portions of time. To conceive them to 
move from their places is to conceive a place to move 
away from itself. 

But as there is nothing to distinguish one portion of 
time from another except the different events which 
occur in them, so there is nothing to distinguish one 
part of space from another except its relation to the 
place of material bodies. We cannot describe the time 
of an event except by reference to some other event, or 
the place of a body except by reference to some other 
body. All our knowledge, both of time and place, is 
essentially relative*. When a man has acquired the 
habit of putting words together, without troubling 
himself to form the thoughts which ought to correspond 
to them, it is easy for him to frame an antithesis between 
this relative knowledge and a so-called absolute know- 
ledge, and to point out our ignorance of the absolute 
position of a point as an instance of the limitation of our 
faculties. Any one, however, who will try to imagine 
the state of a mind conscious of knowing the absolute 
position of a point will ever after be content with our 
relative knowledge. 

* The position seems to be that our knowledge is relative, but 
needs definite space and time as a frame for its coherent ex- 
pression. 



GENERAL MAXIM 13 



19. STATEMENT OF THE GENERAL MAXIM OF 
PHYSICAL SCIENCE 

There is a maxim which is often quoted, that "The 
same causes will always produce the same effects." 

To make this maxim intelligible we must define 
what we mean by the same causes and the same effects, 
since it is manifest that no event ever happens more 
than once, so that the causes and effects cannot be 
the same in all respects. What is really meant is that 
if the causes differ only as regards the absolute time 
or the absolute place at which the event occurs, so 
likewise will the effects. 

The following statement, which is equivalent to the 
above maxim, appears to be more definite, more ex- 
plicitly connected with the ideas of space and time, and 
more capable of application to particular cases : 

"The difference between one event and another does 
not depend on the mere difference of the times or the 
places at which they occur, but only on differences in 
the nature, configuration, or motion of the bodies con- 
cerned." 

It follows from this, that if an event has occurred at 
a given time and place it is possible for an event exactly 
similar to occur at any other time and place. 

There is another maxim which must not be con- 
founded with that quoted at the beginning of this 
article, which asserts "That like causes produce like 
effects." 

This is only true when small variations in the initial 
circumstances produce only small variations in the final 
state of the system*. In a great many physical pheno- 
mena this condition is satisfied; but there are other 

* This implies that it is only in so far as stability subsists that 
principles of natural law can be formulated : it thus perhaps puts 
a limitation on any postulate of universal physical determinacy 
such as Laplace was credited with. 



i 4 INTRODUCTION [CH. i 

cases in which a small initial variation may produce a 
very great change in the final state of the system, as 
when the displacement of the " points " causes a railway 
train to run into another instead of keeping its proper 
course*. 

* We may perhaps say that the observable regularities of 
nature belong to statistical molecular phenomena which have 
settled down into permanent stable conditions. In so far as the 
weather may be due to an unlimited assemblage of local in- 
stabilities, it may not be amenable to a finite scheme of law at all. 



CHAPTER II 



ON MOTION 

20. DEFINITION OF DISPLACEMENT 
WE have already compared the position of different 
points of a system at the same instant of time. We have 
next to compare the position of a point at a given instant 
with its position at a former instant, called the Epoch. 

The vector which indicates the final position of a 
point with respect to its position at the epoch is called 
the Displacement of that point. Thus if A 1 is the initial 
and A 2 the final position of the point A, the line AA Z is 
the displacement of A, and any vector oa drawn from 
the origin o parallel and equal to A ^ indicates this dis- 
placement. 

21. DIAGRAM OF DISPLACEMENT 

If another point of the system is displaced from B to 
B 2 the vector ob paral- 



lel and equal to 
indicates the displace- 
ment of B. 

In like manner the 
displacement of any 
number of points may 
be represented by vec- 
tors drawn from the 
same origin o. This 
system of vectors is 
called the Diagram of 
Displacement. It is 
not necessary to draw 
actual lines to represent 
these vectors ; it is suffi- 
cient to indicate the 




Fig. 3. 



points a, b, etc., at the extremities of the vectors. The 



1 6 ON MOTION [CH. 

diagram of displacement may therefore be regarded as 
consisting of a number of points, , 6, etc., correspond- 
ing with the material particles, A, B, etc., belonging 
to the system, together with a point o, the position of 
which is arbitrary, and which is the assumed origin of 
all the vectors. 

22. RELATIVE DISPLACEMENT 

The line ab in the diagram of displacement repre- 
sents the displacement of the point B with respect 
to A. 

For if in the diagram of displacement (fig. 3) we 
draw ak parallel and equal to B^A^ and in the same 
direction, and join kb, it is easy to show that kb is 
equal and parallel to A 2 B 2 . 

For^the vector kb is the sum of the vectors ka, ao, 
and ob, and A 2 B 2 is_the sum of A 2 A t , A^B lt and 
B t B 2 . But of these ka is the same as A-^B^ ao is the 
same as A 2 A ly and ob is the same as B^B 2t and by 
Article 10 the order of summation is indifferent, so 
that the vector kb is the same, in direction and magni- 
tude, as A 2 B 2 . Now ka or A^ represents the original 
position of B with respect to A, and kb or A 2 B 2 
represents the final position of B with respect to A. 
Hence ab represents the displacement of B with respect 
to A, which was to be proved. 

In Article 20 we purposely omitted to say whether 
the origin to which the original configuration was 
referred, and that to which the final configuration is 
referred, are absolutely the same point, or whether, 
during the displacement of the system, the origin also 
is displaced. 

We may now, for the sake of argument, suppose that 
the origin is absolutely fixed, and that the displace- 
ments represented by oa t ob, etc., are the absolute dis- 
placements. To pass from this case to that in which 



n] DISPLACEMENT 17 

the origin is displaced we have only to take A, one of 
the movable points, as origin. The absolute displace- 
ment of A being represented by oa, the displacement 
of B with respect to A is represented, as we have seen, 
by ab, and so on for any other points of the system. 

The arrangement of the points a, b, etc., in the dia- 
gram of displacement is therefore the same, whether 
we reckon the displacements with respect to a fixed 
point or a displaced point; the only difference is that 
we adopt a different origin of vectors in the diagram of 
displacement, the rule being that whatever point we 
take, whether fixed or moving, for the origin of the 
diagram of configuration, we take the corresponding 
point as origin in the diagram of displacement. If we 
wish to indicate the fact that we are entirely ignorant 
of the absolute displacement in space of any point of 
the system, we may do so by constructing the diagram 
of displacement as a mere system of points, without 
indicating in any way which of them we take as the 
origin. 

This diagram of displacement (without an origin) 
will then represent neither more nor less than all we 
can ever know about the displacement of the system. 
It consists simply of a number of points, a, b, c, etc., 
corresponding to the points A, B, C, etc., of the material 
system, and a vector, as ab represents the displacement 
of B with respect to A. 

23. UNIFORM 1 DISPLACEMENT 

When the displacements of all points of a material 
system with respect to an external point are the same 
in direction and magnitude, the diagram of displace- 
ment is reduced to two points one corresponding to 
the external point, and the other to each and every point 
of the displaced system. In this case the points of the 

1 When the simultaneous values of a quantity for different 
bodies or places are equal, the quantity is said to be uniformly 
distributed in space. 



i8 ON MOTION [CH. 

system are not displaced with respect to one another, 
but only with respect to the external point. 

This is the kind of displacement which occurs when 
a body of invariable form moves parallel to itself. It 
may be called uniform displacement. 

24. ON MOTION 

When the change of configuration of a system is 
considered with respect only to its state at the beginning 
and the end of the process of change, and without 
reference to the time during which it takes place, it is 
called the displacement of the system. 

When we turn our attention to the process of change 
itself, as taking place during a certain time and in a 
continuous manner, the change of configuration is 
ascribed to the motion of the system. 

25. ON THE CONTINUITY OF MOTION 
When a material particle is displaced so as to pass 
from one position to another, it can only do so by 
travelling along some course or path from the one 
position to the other. 

At any instant during the motion the particle will be 
found at some one point 

p f^ II of the path, and if we se- 
^^^ lect any point of the path, 

j\ /r S~"lT^ t ^ ie P art ^ c ^ e W *U P ass t ^ iat 
\_) . " D point once at least 1 during 

its motion. 

This is what is meant 

by saying that the particle describes a continuous path. 
The motion of a material particle which has continuous 
existence in time and space is the type and exemplar 
of every form of continuity. 

1 If the path cuts itself so as to form a loop, as P, Q, R (fig. 4), 
the particle will pass the point of intersection, Q, twice, and if 
the particle returns on its own path, as in the path A , B, C, D, it 
may pass the same point, S, three or more times. 



n] VELOCITY 19 

26. ON CONSTANT 1 VELOCITY 

If the motion of a particle is such that in equal 
intervals of time, however short, the displacements of 
the particle are equal and in the same direction, the 
particle is said to move with constant velocity. 

It is manifest that in this case the path of the body 
will be a straight line, and the length of any part of the 
path will be proportional to the time of describing it. 

The rate or speed of the motion is called the velocity 
of the particle, and its magnitude is expressed by saying 
that it is such a distance in such a time, as, for instance, 
ten miles an hour, or one metre per second. In general 
we select a unit of time, such as a second, and measure 
velocity by the distance described in unit of time. 

If one metre be described in a second and if the 
velocity be constant, a thousandth or a millionth of a 
metre will be described in a thousandth or a millionth 
of a second. Hence, if we can observe or calculate the 
displacement during any interval of time, however short, 
we may deduce the distance which would be described 
in a longer time with the same velocity. This result, 
which enables us to state the velocity during the short 
interval of time, does not depend on the body's actually 
continuing to move at the same rate during the longer 
time. Thus we may know that a body is moving at 
the rate of ten miles an hour, though its motion 
at this rate may last for only the hundredth of a 
second. 

27. ON THE MEASUREMENT OF VELOCITY 
WHEN VARIABLE 

When the velocity of a particle is not constant, its 
value at any given instant is measured by the distance 
which would be described in unit of time by a body 
having the same velocity as that which the particle has 
at that instant. 

1 When the successive values of a quantity for successive 
instants of time are equal, the quantity is said to be constant. 



20 ON MOTION [CH. 

Thus when we say that at a given instant, say one 
second after a body has begun to fall, its velocity is 980 
centimetres per second, we mean that if the velocity of 
a particle were constant and equal to that of the falling 
body at the given instant, it would describe 980 centi- 
metres in a second. 

It is specially important to understand what is meant 
by the velocity or rate of motion of a body, because the 
ideas which are suggested to our minds by considering 
the motion of a particle are those which Newton made 
use of in his method of Fluxions 1 , and they lie at the 
foundation of the great extension of exact science which 
has taken place in modern times. 

28. DIAGRAM OF VELOCITIES 

If the velocity of each of the bodies in the system is 
constant, and if we compare the configurations of the 
system at an interval of a unit of time, then the displace- 
ments, being those produced in unit of time in bodies 
moving with constant velocities, will represent those 
velocities according to the method of measurement 
described in Article 26. 

If the velocities do not actually continue constant 
for a unit of time, then we must imagine another system 
consisting of the same number of bodies, and in which 
the velocities are the same as those of the corresponding 
bodies of the system at the given instant, but remain 
constant for a unit of time. The displacements of this 
system represent the velocities of the actual system at 
the given instant. 

Another mode of obtaining the diagram of velocities 
of a system at a given instant is to take a small interval 
of time, say the nth part of the unit of time, so that 
the middle of this interval corresponds to the given 

1 According to the method of Fluxions, when the value of one 
quantity depends on that of another, the rate of variation of the 
first quantity with respect to the second may be expressed as a 
velocity, by imagining the first quantity to represent the displace- 
ment of a particle, while the second flows uniformly with the time. 



n] DIAGRAM OF VELOCITIES 21 

instant. Take the diagram of displacement corre- 
sponding to this interval and magnify all its dimensions 
n times. The result will be a diagram of the mean 
velocities of the system during the interval. If we now 
suppose the number n to increase without limit the 
interval will diminish without limit, and the mean 
velocities will approximate without limit to the actual 
velocities at the given instant. Finally, when n becomes 
infinite the diagram will represent accurately the velo- 
cities at the given instant. 

29. PROPERTIES OF THE DIAGRAM OF VELOCITIES (fig. 5) 

The diagram of velocities for a system consisting of 
a number of material particles consists of a number 
of points, each corresponding to one of the particles. 



DIAGRAM OF CONFIGURATION. 



DIAGRAM OF VELOCITIES. 



Fig. 5- 

The velocity of any particle B with respect to any 
other, A, is represented in direction and magnitude by 
the line ab in the diagram of velocities, drawn from the 
point a, corresponding to A, to the point 6, corresponding 
to B. 

We may in this way find, by means of the diagram, 
the relative velocity of any two particles. The diagram 
tells us nothing about the absolute velocity of any 
point; it expresses exactly what we can know about 
the motion and no more. If we choose to imagine that 



22 ON MOTION [CH. 

oa represents the absolute velocity of A, then the 
absolute velocity of any other particle, B, will be repre- 
sented by the vector ob, drawn from o as origin to the 
point b, which corresponds to B. 

But as it is impossible to define the position of a 
body except with respect to the position of some point 
of reference, so it is impossible to define the velocity 
of a body, except with respect to the velocity of the 
point of reference. The phrase absolute velocity has 
as little meaning as absolute position. It is better, 
therefore, not to distinguish any point in the diagram 
of velocities as the origin, but to regard the diagram as 
expressing the relations of all the velocities without 
defining the absolute value of any one of them. 

30. MEANING OF THE PHRASE " AT REST " 

It is true that when we say that a body is at rest we 
use a form of words which appears to assert something 
about that body considered in itself, and we might 
imagine that the velocity of another body, if reckoned 
with respect to a body at rest, would be its true and 
only absolute velocity. But the phrase "at rest" 
means in ordinary language "having no velocity with 
respect to that on which the body stands," as, for 
instance, the surface of the earth or the deck of a ship. 
It cannot be made to mean more than this. 

It is therefore unscientific to distinguish between 
rest and motion, as between two different states of a 
body in itself, since it is impossible to speak of a body 
being at rest or in motion except with reference, ex- 
pressed or implied, to some other body. 

3 1 . ON CHANGE OF VELOCITY 

As we have compared the velocities of different 
bodies at the same time, so we may compare the 
relative velocity of one body with respect to another at 
different times. 



n] ACCELERATION 23 

If flj, b lt c^ be the diagram of velocities of the system 
of bodies A, B, C, in its original state, and if a 2 , b 2 , c 2 , 
be the diagram of velocities in the final state of the 
system, then if we take 

any point o>jis origin a 2* 

and draw coa equal QI * 

and parallel to 1 a 2 , b z 

o>j3 equal and parallel b^ 

to 6^2, coy equal and C 2 " 

parallel to c^c^ and so 
on, we shall form a 
diagram of points a, 
j3, y, etc., such that 
any line aft in this w< P* 

diagram represents in 7 

direction and magni- Fig. 6. 

tude the change of the 

velocity of B with respect to A. This diagram may be 
called the diagram of Total Accelerations. 



32. ON ACCELERATION 

The word Acceleration is here used to denote any 
change in the velocity, whether that change be an in- 
crease, a diminution, or a change of direction. Hence, 
instead of distinguishing, as in ordinary language, 
between the acceleration, the retardation, and the 
deflexion of the motion of a body, we say that the 
acceleration may be in the direction of motion, in the 
contrary direction, or transverse to that direction. 

As the displacement of a system is defined to be the 
change of the configuration of the system, so the Total 
Acceleration of the system is defined to be the change of 
the velocities of the system. The process of constructing 
the diagram of total accelerations by a comparison of 
the initial and final diagrams of velocities is the same 



24 ON MOTION [CH. 

as that by which the diagram of displacement was 
constructed by a comparison of the initial and final 
diagrams of configuration. 



33. ON THE RATE OF ACCELERATION 

We have hitherto been considering the total accelera- 
tion which takes place during a certain interval of 
time. If the rate of acceleration is constant, it is 
measured by the total acceleration in a unit of time. 
If the rate of acceleration is variable, its value at a 
given instant is measured by the total acceleration 
in unit of time of a point whose acceleration is 
constant and equal to that of the particle at the given 
instant. 

It appears from this definition that the method of 
deducing the rate of acceleration from a knowledge of 
the total acceleration in any given time is precisely 
analogous to that by which the velocity at any instant 
is deduced from a knowledge of the displacement in 
any given time. 

The diagram of total accelerations constructed for an 
interval of the nth part of the unit of time, and then 
magnified n times, is a diagram of the mean rates of 
acceleration during that interval, and by taking the 
interval smaller and smaller, we ultimately arrive at 
the true rate of acceleration at the middle of that 
interval. 

As rates of acceleration have to be considered in 
physical science much more frequently than total ac- 
celerations, the word acceleration has come to be 
employed in the sense in which we have hitherto used 
the phrase rate of acceleration. 

In future, therefore, when we use the word accelera- 
tion without qualification, we mean what we have here 
described as the rate of acceleration. 



n] ACCELERATION 25 

34. DIAGRAM OF ACCELERATIONS 
The diagram of accelerations is a system of points, 
each of which corresponds to one of the bodies of the 
material system, and is such that any line aft in the 
diagram represents the rate of acceleration of the body 
B with respect to the body A. 

It may be well to observe here that in the diagram 
of configuration we use the capital letters, A, B, C, etc., 
to indicate the relative position of the bodies of the 
system ; in the diagram of velocities we use the small 
letters, a, b, c, etc., to indicate the relative velocities of 
these bodies ; and in the diagram of accelerations we use 
the Greek letters, a, j8, y, etc., to indicate their relative 
accelerations. 

35. ACCELERATION A RELATIVE TERM 
Acceleration, like position and velocity, is a relative 
term and cannot be interpreted absolutely*. 

If every particle of the material universe within the 
reach of our means of observation were at a given 
instant to have its velocity altered by compounding 
therewith a new velocity, the same in magnitude and 
direction for every such particle, all the relative motions 
of bodies within the system would go on in a perfectly 
continuous manner, and neither astronomers nor 
physicists, though using their instruments all the 
while, would be able to find out that anything had 
happenedf . 

It is only if the change of motion occurs in a different 
manner in the different bodies of the system that any 
event capable of being observed takes place. 

* A noteworthy case of relativity is Euler's investigation of the 
motion of a solid body as specified with reference to its own 
succession of instantaneous positions. 

f This appears to be a very drastic postulate of relativity: 
a universal imposed acceleration can have no effect during its 
occurrence only when all applied forces are proportional to mass. 
See Appendix I. 



CHAPTER III 
ON FORCE 

36. KINEMATICS AND KINETICS 

WE have hitherto been considering the motion of a 
system in its purely geometrical aspect. We have 
shown how to study and describe the motion of such a 
system, however arbitrary, without taking into account 
any of the conditions of motion which arise from the 
mutual action between the bodies. 

The theory of motion treated in this way is called 
Kinematics. When the mutual action between bodies 
is taken into account, the science of motion is called 
Kinetics, and when special attention is paid to force as 
the cause of motion, it is called Dynamics. 

37. MUTUAL ACTION BETWEEN Two BODIES STRESS 
The mutual action between two portions of matter 
receives different names according to the aspect under 
which it is studied, and this aspect depends on the 
extent of the material system which forms the subject 
of our attention. 

If we take into account the whole phenomenon of the 
action between the two portions of matter, we call it 
Stress. This stress, according to the mode in which it 
acts, may be described as Attraction, Repulsion, Ten- 
sion, Pressure, Shearing stress, Torsion, etc. 

38. EXTERNAL FORCE 

But if, as in Article 2, we confine our attention to 
one of the portions of matter, we see, as it were, only 
one side of the transaction namely, that which affects 
the portion of matter under our consideration and we 
call this aspect of the phenomenon, with reference to 
its effect, an External Force acting on that portion of 



CH. in] LAWS OF MOTION 27 

matter, and with reference to its cause we call it the 
Action of the other portion of matter. The opposite 
aspect of the stress is called the Reaction on the other 
portion of matter. 

39. DIFFERENT ASPECTS OF THE SAME PHENOMENON 

In commercial affairs the same transaction between 
two parties is called Buying when we consider one 
party, Selling when we consider the other, and Trade 
when we take both parties into consideration. 

The accountant who examines the records of the 
transaction finds that the two parties have entered it on 
opposite sides of their respective ledgers, and in com- 
paring the books he must in every case bear in mind in 
whose interest each book is made up. 

For similar reasons in dynamical investigations we 
must always remember which of the two bodies we are 
dealing with, so that we may state the forces in the 
interest of that body, and not set down any of the forces 
on the wrong side of the account. 

40. NEWTON'S LAWS OF MOTION 

External or "impressed " force considered with refer- 
ence to its effect* namely, the alteration of the motions 
of bodies is completely defined and described in 
Newton's three laws of motion. 

The first law tells us under what conditions there is 
no external force. 

The second shows us how to measure the force when 
it exists. 

The third compares the two aspects of the action 
between two bodies, as it affects the one body or the 
other.. 

* As to its nature, a stress, or balanced set of forces, is deter- 
mined by the alteration of the permanent configuration of the 
bodies concerned, which reveals its existence and forms the basis 
of its statical measure; or else by some other property of matter. 
Cf. Art. 68. 



28 ON FORCE [CH. 

41. THE FIRST* LAW OF MOTION 
Law I. Every body perseveres in its state of rest or 
of moving uniformly in a straight line,- except in so 
far as it is made to change that state by external forces . 

The experimental argument for the truth of this 
law is, that in every case in which we find an alteration 
of the state of motion of a body, we can trace this 
alteration to some action between that body and another, 
that is to say, to an external force. The existence of 
this action is indicated by its effect on the other 
body when the motion of that body can be observed. 
Thus the motion of a cannon ball is retarded, but 
this arises from an action between the projectile and 
the air which surrounds it, whereby the ball experiences 
a force in the direction opposite to its relative motion, 
while the air, pushed forward by an equal force, is 
itself set in motion, and constitutes what is called the 
wind of the cannon ball. 

But our conviction of the truth of this law may be 
greatly strengthened by considering what is involved in 
a denial of it. Given a body in motion. At a given 
instant let it be left to itself and not acted on by any 
force. What will happen? According to Newton's 
law it will persevere in moving uniformly in a straight 
line, that is, its velocity will remain constant both in 
direction and magnitude. 

If the velocity does not remain constant let us 
suppose it to vary. The change of velocity, as we saw in 
Article 31, must have a definite direction and magni- 
tude. By the maxim of Article 19 this variation must 
be the same whatever be the time or place of the 
experiment. The direction of the change ef motion 
must therefore be determined either by the direction of 
the motion itself, or by some direction fixed in the 
body. 

Let us, in the first place, suppose the law to be that 
the velocity diminishes at a certain rate, which for the 



in] FIRST LAW OF MOTION 29 

sake of the argument we may suppose so slow that by 
no experiments on moving bodies could we have 
detected the diminution of velocity in hundreds of 
years. 

The velocity referred to in this hypothetical law can 
only be the velocity referred to a point absolutely at 
rest. For if it is a relative velocity its direction as 
well as its magnitude depends on the velocity of the 
point of reference. 

If, when referred to a certain point, the body appears 
to be moving northward with diminishing velocity, we 
have only to refer it to another point moving northward 
with a uniform velocity greater than that of the body, 
and it will appear to be moving southward with in- 
creasing velocity. 

Hence the hypothetical law is without meaning, un- 
less we admit the possibility of defining absolute rest 
and absolute velocity*. 

Even if we admit this as a possibility, the hypothetical 
law, if found to be true, might be interpreted, not as 
a contradiction of Newton's law, but as evidence of 
the resisting action of some medium in space. 

To take another case. Suppose the law to be that a 
body, not acted on by any force, ceases at once to move. 
This is not only contradicted by experience, but it leads 
to a definition of absolute rest as the state which a body 
assumes as soon as it is freed from the action of ex- 
ternal forces. 

It may thus be shown that the denial of Newton's 
law is in contradiction to the only system of consistent 
doctrine about space and time which the human mind 
has been able to formf. 

* An aether might do this. But even in Maxwell's aether an 
isolated body losing energy by radiation would suffer no change 
of velocity thereby. 

f The argument of this section may be made more definite. 
It is a result of observation that the more isolated a body is from 
the influence of other bodies, the more nearly is its velocity 
constant with reference to an assignable frame of reference. A 



30 ON FORCE [CH. 

42. ON THE EQUILIBRIUM OF FORCES 
If a body moves with constant velocity in a straight 
line, the external forces, if any, which act on it, balance 
each other, or are in equilibrium. 

Thus if a carriage in a railway train moves with 
constant velocity in a straight line, the external forces 
which act on it such as the traction of the carriage in 
front of it pulling it forwards, the drag of that behind 
it, the friction of the rails, the resistance of the air 
acting backwards, the weight of the carriage acting 
downwards, and the pressure of the rails acting up- 
wards must exactly balance each other. 

Bodies at rest with respect to the surface of the earth 
are really in motion, and their motion is not constant nor 
in a straight line. Hence the forces which act on them 
are not exactly balanced. The apparent weight of bodies 
is estimated by the upward force required to keep them 
at rest relatively to the earth. The apparent weight is 

main problem of physical dynamics is to determine with in- 
creasing approximation a frame for which this principle holds, 
for all systems, with the greatest attainable precision. A frame 
of space and time thus determined has been called (after James 
Thomson) a frame of inertia. The statements in the text can be 
reconstructed with regard to a reference frame which is a frame 
of inertia. But given one frame of inertia, any other frame moving 
with any uniform translatory velocity with respect to it, is also 
a frame of inertia. Thus a first approximation for local purposes 
to a frame of inertia is one fixed with reference to the surrounding 
landscape; when the range of phenomena is widened, astronomers 
have to change to a frame containing the axis of the earth's 
diurnal rotation, and involving a definite value for the length of 
the sidereal day : this again has to be corrected for the very slow 
movement of the earth's axis that is revealed by the Precession 
of the Equinoxes: and so on. Such a frame of inertia represents 
in practical essentials the Newtonian absolute space and time: 
it is the simplest and most natural scheme of mapping an ex- 
tension into which dynamical phenomena can be fitted. If we 
assume that space is occupied by a uniform static aether through 
whose mediation influences are transmitted from one material 
body to another, the properties of that medium will afford unique 
specification of an absolute space and time having physical 
properties as well as relations of extension. See Appendix I. 



in] EQUILIBRIUM 31 

therefore rather less than the attraction of the earth, 
and makes a smaller angle with the axis of the earth, 
so that the combined effect of the supporting force and 
the earth's attraction is a force perpendicular to the 
earth's axis just sufficient to cause the body to keep 
to the circular path which it must describe if resting 
on the earth*. 

43. DEFINITION OF EQUAL TIMES 

The first law of motion, by stating under what cir- 
cumstances the velocity of a moving body remains 
constant, supplies us with a method of defining equal 
intervals of time. Let the material system consist of 
two bodies which do not act on one another, and 
which are not acted on by any body external to the 
system If one of these bodies is in motion with respect 
to the other, the relative velocity will, by the first 
law of motion, be constant and in a straight line. 

Hence intervals of time are equal when the relative 
displacements during those intervals are equalf . 

This might at first sight appear to be nothing more 
than a definition of what we mean by equal intervals of 
time, an expression which we have not hitherto defined 
at all. 

But if we suppose another moving system of two 
bodies to exist, each of which is not acted upon by 
any body whatever, this second system will give 
us an independent method of comparing intervals of 
time. 

The statement that equal intervals of time are those 
during which equal displacements occur in any such 

* See end of Appendix I. 

t This statement refers to the displacement of one body 
measured on a complete frame of reference attached to the other. 
It would not be true for two points moving with uniform velocities, 
if relative displacement meant merely change of distance between 
them. In fact their mutual distance undergoes acceleration at a 
rate varying inversely as the cube of that distance : to an observer 
not sensible of directions they would seem to repel each other 
with a force obeying that law of action. 



32 ON FORCE [CH. 

system, is therefore equivalent to the assertion that the 
comparison of intervals of time leads to the same 
result whether we use the first system of two bodies or 
the second system as our time-piece. 

We thus see the theoretical possibility of comparing 
intervals of time however distant, though it is hardly 
necessary to remark that the method cannot be put in 
practice in the neighbourhood of the earth, or any other 
large mass of gravitating matter. 

44. THE SECOND LAW OF MOTION 
Law II. Change of motion is proportional to the 

impressed force, and takes place in the direction in which 

the force is impressed. 

By motion Newton means what in modern scientific 

language is called Momentum, in which the quantity of 

matter moved is taken into account as well as the rate 

at which it travels. 

By impressed force he means what is now called 

Impulse, in which the time during which the force acts 

is taken into account as well as the intensity of the force. 

45. DEFINITION OF EQUAL MASSES AND OF 
EQUAL FORCES 

An exposition of the law therefore involves a defini- 
tion of equal quantities of matter and of equal forces. 

We shall assume that it is possible to cause the force 
with which one body acts on another to be of the same 
intensity on different occasions. 

If we admit the permanency of the properties of bodies 
this can be done. We know that a thread of caoutchouc 
when stretched beyond a certain length exerts a tension 
which increases the more the thread is elongated. On 
account of this property the thread is said to be elastic. 
When the same thread is drawn out to the same length 
it will, if its properties remain constant, exert the same 
tension. Now let one end of the thread be fastened to 



m] MASS 33 

a body, M, not acted on by any other force than the 
tension of the thread, and let the other end be held 
in the hand and pulled in a constant direction with a 
force just sufficient to elongate the thread to a given 
length. The force acting on the body will then be of 
a given intensity, F. The body will acquire velocity, 
and at the end of a unit of time this velocity will have 
a certain value, V . 

If the same string be fastened to another body, N, 
and pulled as in the former case, so that the elongation 
is the same as before, the force acting on the body 
will be the same, and if the velocity communicated to 
N in a unit of time is also the same, namely V, then 
we say of the two bodies M and N that they consist 
of equal quantities of matter, or, in modern language, 
they are equal in mass. In this way, by the use of an 
elastic string, we might adjust the masses of a number 
of bodies so as to be each equal to a standard unit 
of mass, such as a pound avoirdupois, which is the 
standard of mass in Britain. 

46. MEASUREMENT OF MASS 

The scientific value of the dynamical method of com- 
paring quantities of matter is best seen by comparing it 
with other methods in actual use. 

As long as we have to do with bodies of exactly the 
same kind, there is no difficulty in understanding how 
the quantity of matter is to be measured. If equal 
quantities of the substance produce equal effects of any 
kind, we may employ these effects as measures of the 
quantity of the substance. 

For instance, if we are dealing with sulphuric acid of 
uniform strength, we may estimate the quantity of a 
given portion of it in several different ways. We may 
weigh it, we may pour it into a graduated vessel, and 
so measure its volume, or we may ascertain how much 
of a standard solution of potash it will neutralise. 

We might use the same methods to estimate a 



34 ON FORCE [CH. 

quantity of nitric acid if we were dealing only with 
nitric acid; but if we wished to compare a quantity 
of nitric acid with a quantity of sulphuric acid we 
should obtain different results by weighing, by mea- 
suring, and by testing with an alkaline solution. 

Of these three methods, that of weighing depends on 
the attraction between the acid and the earth, that of 
measuring depends on the volume which the acid 
occupies, and that of titration depends on its power of 
combining with potash. 

In abstract dynamics, however, matter is considered 
under no other aspect than as that which can have its 
motion changed by the application of force. Hence 
any two bodies are of equal mass if equal forces applied 
to these bodies produce, in equal times, equal changes 
of velocity. This is the only definition of equal masses 
which can be admitted in dynamics, and it is applicable 
to all material bodies, whatever they may be made of. 

It is an observed fact that bodies of equal mass, 
placed in the same position relative to the earth, are 
attracted equally towards the earth, whatever they are 
made of; but this is not a doctrine of abstract dynamics, 
founded on axiomatic principles, but a fact discovered 
by observation, and verified by the careful experiments 
of Newton*, on the times of oscillation of hollow wooden 
balls suspended by strings of the same length, and con- 
taining gold, silver, lead, glass, sand, common salt, 
wood, water, and wheat. 

The fact, however, that in the same geographical 
position the weights of equal masses are equal, is so 
well established, that no other mode of comparing 
masses than that of comparing their weights is ever 
made use of, either in commerce or in science, except 
in researches undertaken for the special purpose of 

* Principia, III. Prop. 6. Actual weight is a compound effect, 
in the main attraction, but diminished by reaction against 
centripetal acceleration of the mass due to the earth's rotation. 
See p. 143. 



in] MEASUREMENT 35 

determining in absolute measure the weight of unit of 
mass at different parts of the earth's surface. The 
method employed in these researches is essentially the 
same as that of Newton, namely, by measuring the 
length of a pendulum which swings seconds. 

The unit of mass in this country is defined by the 
Act of Parliament (18 & 19 Viet. c. 72, July 30, 1855) 
to be a piece of platinum marked "P.S., 1844, i Ib." 
deposited in the office of the Exchequer, which "shall 
be and be denominated the Imperial Standard Pound 
Avoirdupois." One seven-thousandth part of this 
pound is a grain.. The French standard of mass is the 
"Kilogramme des Archives," made of platinum by 
Borda. Professor Miller finds the kilogramme equal to 
1 5432-34 8 74 grains. 

47. NUMERICAL MEASUREMENT OF FORCE 

The unit of force is that force which, acting on the 
unit of mass for the unit of time, generates unit of 
velocity. 

Thus the weight of a gramme that is to say, the 
force which causes it to fall may be ascertained by 
letting it fall freely. At the end of one second its 
velocity will be about 981 centimetres per second if the 
experiment be in Britain. Hence the weight of a gramme 
is represented by the number 981, if the centimetre, 
the gramme, and the second are taken as the funda- 
mental units. 

It is sometimes convenient to compare forces with 
the weight of a body, and to speak of a force of so 
many pounds weight or grammes weight. This is 
called Gravitation measure. We must remember, how- 
ever, that though a pound or a gramme is the same all 
over the world, the weight of a pound or a gramme is 
greater in high latitudes than near the equator, and 
therefore a measurement of force in gravitation measure 
is of no scientific value unless it is stated in what part 
of the world the measurement was made. 

32 



36 ON FORCE [CH. 

If, as in Britain, the units of length, mass, and time 
are one foot, one pound, and one second, the unit of 
force is that which, in one second, would communicate 
to one pound a velocity of one foot per second. This 
unit of force is called a Poundal. 

In the French metric system the units are one 
centimetre, one gramme, and one second. The force 
which in one second would communicate to one gramme 
a velocity of one centimetre per second is called a Dyne. 

Since the foot is 30-4797 centimetres and the pound 
is 453-59 grammes, the poundal is 13825-38 dynes. 



48. SIMULTANEOUS ACTION OF FORCES ON A BODY 

Now let a unit of force act for unit of time upon unit 
of mass. The velocity of the mass will be changed, 
and the total acceleration will be unity in the direction 
of the force. 

The magnitude and direction of this total acceleration 
will be the same whether the body is originally at rest 
or in motion*. For the expression "at rest" has no 
scientific meaning, and the expression " in motion," if it 
refers to relative motion, may mean anything, and if it 
refers to absolute motion can only refer to some medium 
fixed in space. To discover the existence of a medium, 
and to determine our velocity with respect to it by 
observation on the motion of bodies, is a legitimate 
scientific inquiry, but supposing all this done we should 
have discovered, not an error in the laws of motion, 
but a new fact in science. 

Hence the effect of a given force on a body does not 
depend on the motion of that body. 

Neither is it affected by the simultaneous action of 
other forces on the body. For the effect of these 
forces on the body is only to produce motion in the 
body, and this does not affect the acceleration produced 
by the first force. 

* Cf. Appendix I. 



in] IMPULSE 37 

Hence we arrive at the following form of the law. 
When any number of forces act on a body, the accelera- 
tion due to each force is the same in direction and magnitude 
as if the others had not been in action. 

When a force, constant in direction and magnitude, 
acts on a body, the total acceleration is proportional to 
the interval of time during which the force acts. 

For if the force produces a certain total acceleration 
in a given interval of time, it will produce an equal 
total acceleration in the next, because the effect of the 
force does not depend upon the velocity which the 
body has when the force acts on it. Hence in every 
equal interval of time there will be an equal change of 
the velocity, and the total change of velocity from the 
beginning of the motion will be proportional to the time 
of action of the force. 

The total acceleration in a given time is proportional 
to the force. 

For if several equal forces act in the same direction 
on the same body in the same direction, each produces 
its effect independently of the others. Hence the total 
acceleration is proportional to the number of the equal 
forces. 

49. ON IMPULSE 

The total effect of a force in communicating velocity 
to a body is therefore proportional to the force and to 
the time during which it acts conjointly. 

The product of the time of action of a force into its 
intensity if it is constant, or its mean intensity if it is 
variable, is called the Impulse of the force. 

There are certain cases in which a force acts for so 
short a time that it is difficult to estimate either its 
intensity or the time during which it acts. But it is 
comparatively easy to measure the effect of the force 
in altering the motion of the body on which it acts, 
which, as we have seen, depends on the impulse. 

The word impulse was originally used to denote the 



38 ON FORCE [CH. 

effect of a force of short duration, such as that of a 
hammer striking a nail. There is no essential differ- 
ence, however, between this case and any other case 
of the action of force. We shall therefore use the 
word impulse as above defined, without restricting it 
to cases in which the action is of an exceptionally 
transient character. 

50. RELATION BETWEEN FORCE AND MASS 
If a force acts on a unit of mass for a certain interval 
of time, the impulse, as we have seen, is measured 
by the velocity generated. 

If a number of equal forces act in the same direction, 
each on a unit of mass, the different masses will all 
move in the same manner, and may be joined together 
into one body without altering the phenomenon. The 
velocity of the whole body is equal to that produced by 
one of the forces acting on a unit of mass. 

Hence the force required to produce a given change 
of velocity in a given time is proportional to the 
number of units of mass* of which the body consists. 

51. ON MOMENTUM 

The numerical value of the Momentum of a body is 
the product of the number of units of mass in the body 
into the number of units of velocity with which it is 
moving. 

The momentum of any body is thus measured in 
terms of the momentum of unit of mass moving with 
unit of velocity, which is taken as the unit of momentum. 

The direction of the momentum is the same as that 
of the velocity, and as the velocity can only be estimated 
with respect to some point of reference, so the particular 
value of the momentum depends on the point of refer- 

* Here mass means the measure of the inertia rather than the 
quantity of matter; at extremely great speeds they would not 
be proportional, but connected by a law involving the speed, so 
that momentum or impulse would then be the primary quantity 
and inertia a derived one. 



in] SECOND LAW OF MOTION 39 

ence which we assume. The momentum of the moon, 
for example, will be very different according as we take 
the earth or the sun for the point of reference. 

52. STATEMENT OF THE SECOND LAW OF MOTION IN 

TERMS OF IMPULSE AND MOMENTUM 
The change of momentum of a body is numerically equal 
to the impulse which produces it, and is in the same 
direction. 

53. ADDITION OF FORCES 

If any number of forces act simultaneously on a 
body, each force produces an acceleration proportional 
to its own magnitude (Article 48). Hence if in the 
diagram of accelerations (Article 34) we draw from 
any origin a line representing in direction and magni- 
tude the acceleration due to one of the forces, and 
from the end of this line another representing the ac- 
celeration due to another force, and so on, drawing lines 
for each of the forces taken in any order, then the line 
drawn from the origin to the extremity of the last of the 
lines will represent the acceleration due to the combined 
action of all the forces. 

Since in this diagram lines which represent the 
accelerations are in the same proportion as the forces 
to which these accelerations are due, we may consider 
the lines as representing these forces themselves. 
The diagram, thus understood, may be called a Diagram 
of Forces, and the line from the origin to the extremity 
of the series represents the Resultant Force. 

An important case is that in which the set of lines 
representing the forces terminate at the origin so as to 
form a closed figure. In this case there is no resultant 
force, and no acceleration. The effects of the forces are 
exactly balanced, and the case is one of equilibrium. 
The discussion of cases of equilibrium forms the subject 
of the science of Statics. 

It is manifest that since the system of forces is 



40 ON FORCE [CH. 

exactly balanced, and is equivalent to no force at all*, 
the forces will also be balanced if they act in the same 
way on any other material systemf, whatever be the 
mass of that system. This is the reason why the con- 
sideration of mass does not enter into statical investi- 
gations. 

54. THE THIRD LAW OF MOTION 
Law III. Reaction is always equal and opposite to 
action, that is to say, the actions of two bodies upon each 
other are always equal and in opposite directions. 

When the bodies between which the action takes 
place are not acted on by any other force, the changes 
in their respective momenta produced by the action are 
equal and in opposite directions. 

The changes in the velocities of the two bodies are 
also in opposite directions, but not equal, except in the 
case of equal masses. In other cases the changes of 
velocity are in the inverse ratio of the masses. 

55. ACTION AND REACTION ARE THE PARTIAL 
ASPECTS OF A STRESS 

We have already (Article 37) used the word Stress 
to denote the mutual action between two portions of 
matter. This word was borrowed from common 
language, and invested with a precise scientific meaning 
by the late Professor Rankine, to whom we are indebted 
for several other valuable scientific terms. 

As soon as we have formed for ourselves the idea of 
a stress, such as the Tension of a rope or the Pressure 
between two bodies, and have recognised its double 
aspect as it affects the two portions of matter between 

* Except however as regards the strains which the system of 
forces sets up in a deformable body, in cases when they do not 
all act at the same point. It is when these strains are not regarded, 
or the body on which they act is considered as perfectly rigid, that 
we can speak of the statical equivalence of two systems of forces. 

f If the forces do not act at the same point, the system must 
be a rigid one, else it will be deformed by them. 



in] ACTION AND REACTION . 41 

which it acts, the third law of motion is seen to be 
equivalent to the statement that all force is of the nature 
of stress, that stress exists only between two portions 
of matter, and that its effects on these portions of 
matter (measured by the momentum generated in a 
given time) are equal and opposite. 

The stress is measured numerically by the force 
exerted on either of the two portions of matter. It is 
distinguished as a tension when the force acting on 
either portion is towards the other, and as a pressure 
when the force acting on either portion is away from 
the other. I 

When the force is inclined to the surface which 
separates the two portions of matter the stress cannot 
be distinguished by any term in ordinary language, but 
must be defined by technical mathematical terms. 

When a tension is exerted between two bodies by the 
medium of a string, the stress, properly speaking, is 
between any two parts into which the string may be 
supposed to be divided by an imaginary section or 
transverse interface. If, however, we neglect the weight 
of the string, each portion of the string is in equilibrium 
under the action of the tensions at its extremities, so 
that the tensions at any two transverse interfaces of the 
string must be the same. For this reason we often 
speak of the tension of the string as a whole, without 
specifying any particular section of it, and also the 
tension between the two bodies, without considering 
the nature of the string through which the tension is 
exerted. 

56. ATTRACTION AND REPULSION 

There are other cases in which two bodies at a dis- 
tance appear mutually to act on each other, though we 
are not able to detect any intermediate body, like the 
string in the former example, through which the action 
takes place. For instance, two magnets or two electri- 
fied bodies appear to act on each other when placed at 



42 ON FORCE [CH. 

considerable distances apart, and the motions of the 
heavenly bodies are observed to be affected in a manner 
which depends on their relative position. 

This mutual action between distant bodies is called 
attraction when it tends to bring them nearer, and 
repulsion when it tends to separate them. 

In all cases, however, the action and reaction between 
the bodies are equal and opposite. 

57. THE THIRD LAW TRUE OF ACTION AT A DISTANCE 

The fact that a magnet draws iron towards it was 
noticed by the ancients, but no attention was paid to 
the force with which the iron attracts the magnet. 
Newton, however, by placing the magnet in one vessel 
and the iron in another, and floating both vessels in 
water so as to touch each other, showed experimentally 
that as neither vessel was able to propel the other along 
with itself through the water, the attraction of the iron 
on the magnet must be equal and opposite to that of 
the magnet on the iron, both being equal to the pressure 
between the two vessels. 

Having given this experimental illustration Newton 
goes on to point out the consequence of denying the 
truth of this law. For instance, if the attraction of any 
part of the earth, say a mountain, upon the remainder 
of the earth were greater or less than that of the remain- 
der of the earth upon the mountain, there would be a 
residual force, acting upon the system of the earth and 
the mountain as a whole, which would cause it to move 
off, with an ever-increasing velocity, through infinite 
space. 

58. NEWTON'S PROOF NOT EXPERIMENTAL 

This is contrary to the first law of motion, which 
asserts that a body does not change its state of motion 
unless acted on by external force. It cannot be affirmed 
to be contrary to experience, for the effect of an in- 
equality between the attraction of the earth on the 



in] ATTRACTION 43 

mountain and the mountain on the earth would be the 
same as that of a force equal to the difference of these 
attractions acting in the direction of the line joining the 
centre of the earth with the mountain. 

If the mountain were at the equator the earth would 
be made to rotate about an axis parallel to the axis 
about which it would otherwise rotate, but not passing 
exactly through the centre of the earth's mass*. 

If the mountain were at one of the poles, the con- 
stant force parallel to the earth's axis would cause the 
orbit of the earth about the sun to be slightly shifted 
to the north or south of a plane passing through the 
centre of the sun's mass. 

If the mountain were at any other part of the earth's 
surface its effect would be partly of the one kind and 
partly of the other. 

Neither of these effects, unless they were very large, 
could be detected by direct astronomical observations, 
and the indirect method of detecting small forces, by 
their effect in slowly altering the elements of a planet's 
orbit, presupposes that the law of gravitation is known 
to be true. To prove the laws of motion by the law of 
gravitation would be an inversion of scientific order. 
We might as well prove the law of addition of numbers 
by the differential calculus. 

We cannot, therefore, regard Newton's statement as 
an appeal to experience and observation, but rather as 
a deduction of the third law of motion from the first. 

* This is because such a residual force would revolve along with 
the earth's diurnal motion. If F is this force, E the earth's mass 
and u its angular velocity, the altered axis of rotation would be 
at a distance R from the centre of mass such that F Eu 2 R. 

In the next sentence the direction of the residual force is con- 
stant ; and the earth being held in an orbit around the sun by the 
gravitational attraction, that force is transferred to the solar 
system as a whole, to which accordingly, and not to the earth 
alone, the final statement in Art. 57 would apply. 



CHAPTER IV 

ON THE PROPERTIES OF THE CENTRE OF 
MASS OF A MATERIAL SYSTEM 

59. DEFINITION OF A MASS- VECTOR 
WE have seen that a vector represents the operation 
of carrying a tracing point from a given origin to a given 
point. 

Let us define a mass-vector as the operation of carry- 
ing a given mass from the origin to the given point. 
The direction of the mass-vector is the same as that of 
the vector of the mass, but its magnitude is the product 
of the mass into the vector of the mass. 

Thus if OA is the vector of the mass A, the mass- 
vector is OA .A. 

60. CENTRE OF MASS OF Two PARTICLES 
If A and B are two masses, and if a point C be taken 
in the straight line AB, so that BC is to CA as A to B, 
then the mass-vector of a mass A + B placed at C is 
equal to the sum of the mass- vectors of A and B. 

(OC + CB)B 




Now the mass-vectors CA . A and 
CB . B are equal and opposite, and 
so destroy each other, so that 



or, C is a point such that if the 
masses of A and B were concen- 
Fig. 7. trated at C, their mass-vector from 

any origin O would be the same as 
when A and B are in their actual positions. The point 
C is called the Centre of Mass of A and B. 



CH. iv] MOMENTUM 45 

61. CENTRE OF MASS OF A SYSTEM 

If the system consists of any number of particles, we 
may begin by finding the centre of mass of any two 
particles, and substituting for the two particles a particle 
equal to their sum placed at their centre of mass. We 
may then find the centre of mass of this particle, to- 
gether with the third particle of the system, and place 
the sum of the three particles at this point, and so on 
till we have found the centre of mass of the whole 
system. 

The mass-vector drawn from any origin to a mass 
equal to that of the whole system placed at the centre 
of mass of the system is equal to the sum of the mass- 
vectors drawn from the same origin to all the particles 
of the system. 

It follows, from the proof in Article 60, that the 
point found by the construction here given satisfies this 
condition. It is plain from the condition itself that 
only one point can satisfy it. Hence the construction 
must lead to the same result, as to the position of the 
centre of mass, in whatever order we take the particles 
of the system. 

The centre of mass is therefore a definite point in 
the diagram of the configuration of the system. By 
assigning to the different points in the diagrams of 
displacement, velocity, total acceleration, and rate of 
acceleration, the masses of the bodies to which they 
correspond, we may find in each of these diagrams a 
point which corresponds to the centre of mass, and 
indicates the displacement, velocity, total acceleration, 
or rate of acceleration of the centre of mass. 

62. MOMENTUM REPRESENTED AS THE RATE OF 

CHANGE OF A MASS-VECTOR 

In the diagram of velocities, if the points o, a, b, c, 
correspond to the velocities of the origin O and the 
bodies A, B, C, and if p be the centre of mass of A 




46 ON CENTRE OF MASS [CH. 

and B placed at a and b respectively, and if q is the 
centre of mass of A + B placed at p and C at c, then 
^ q will be the centre of mass of 

.fr the system of bodies A, B, C, at 
a, b, c, respectively. 

The velocity of A with respect 
to O is indicated by the vector oa, 
and that of B and C by ob and oc. 
op is the velocity of the centre of 
mass of A and 5, and oq that of 
the centre of mass of A, B, and C, with respect to O. 

The momentum of A with respect to O is the product 
of the velocity into the mass, or oa . A, or what we have 
already called the mass- vector, drawn from o to the 
mass A at a. Similarly the momentum of any other 
body is the mass-vector drawn from o to the point on 
the diagram of velocities corresponding to that body, and 
the momentum of the mass of the system concentrated 
at the centre of mass is the mass-vector drawn from o 
to the whole mass at q. 

Since, therefore, a mass- vector in the diagram of 
velocities is what we have already defined as a momen- 
tum, we may state the property proved in Article 61 
in terms of momenta, thus : The momentum of a mass 
equal to that of the whole system, moving with the 
velocity of the centre of mass of the system, is equal in 
magnitude and parallel in direction to the sum of the 
momenta of all the particles of the system. 

63. EFFECT OF EXTERNAL FORCES ON THE MOTION 
OF THE CENTRE OF MASS 

In the same way in the diagram of Total Acceleration 
the vectors o>a, co/J, etc., drawn from the origin, represent 
the change of velocity of the bodies A, B, etc., during 
a certain_intervaj_of time. The corresponding mass- 
vectors, u>a.A,a>fi.B, etc., represent the correspond- 



iv] ITS MOTION 47 

ing changes of momentum, or, by the second law of 
motion, the impulses of the forces acting on these 
bodies during that interval of 

time. If K is the centre of mass a ^ ]3 

of the system, IDK is the change 

of velocity during the interval, 

and OJK (A + B + C) is the 

momentum generated in the &>. 

mass concentrated at the centre pi g 9 

of gravity. Hence, by Article 

61, the change of momentum of the imaginary mass 

equal to that of the whole system concentrated at the 

centre of mass is equal to the sum of the changes of 

momentum of all the different bodies of the system. 

In virtue of the second law of motion we may put 
this result in the following form: 

The effect of the forces acting on the different bodies 
of the system in altering the motion of the centre of 
mass of the system is the same as if all these forces 
had been applied to a mass equal to the whole mass of 
the system, and coinciding with its centre of mass. 

64. THE MOTION OF THE CENTRE OF MASS OF A 

SYSTEM is NOT AFFECTED BY THE MUTUAL 

ACTION OF THE PARTS OF THE SYSTEM 

For if there is an action between two parts of the 
system, say A and B, the action of A on B is always, 
by the third law of motion, equal and opposite to the 
reaction of B on A. The momentum generated in B 
by the action of A during any interval is therefore 
equal and opposite to that generated in A by the 
reaction of B during the same interval, and the motion 
of the centre of mass of A and B is therefore not 
affected by their mutual action. 

We may apply the result of the last article to this 
case and say, that since the forces on A and on B arising 
from their mutual action are equal and opposite, and 



48 ON CENTRE OF MASS [en. 

since the effect of these forces on the motion of the 
centre of mass of the system is the same as if they had 
been applied to a particle whose mass is equal to the 
whole mass of the system, and since the effect of two 
forces equal and opposite to each other is zero, the 
motion of the centre of mass will not be affected. 

65. FIRST AND SECOND LAWS OF MOTION 
This is a very important result. It enables us to 
render more precise the enunciation of the first and 
second laws of motion, by defining that by the velocity 
of a body is meant the velocity of its centre of mass. The 
body may be rotating, or it may consist of parts, and be 
capable of changes of configuration, so that the motions 
of different parts may be different, but we can still 
assert the laws of motion in the following form : 

Law I. The centre of mass of the system perseveres 
in its state of rest, or of uniform motion in a straight 
line, except in so far as it is made to change that state 
by forces acting on the system from without. 

Law II. The change of momentum* of the system 
during any interval of time is measured by the sum of 
the impulses of the external forces during that interval. 

66. METHOD OF TREATING SYSTEMS OF MOLECULES 
When the system is made up of parts which are so 
small that we cannot observe them, and whose motions 
are so rapid and so variable that even if we could 
observe them we could not describe them, we are 
still able to deal with the motion of the centre of mass 
of the system, because the internal forces which cause 
the variation of the motion of the parts do not affect 
the motion of the centre of mass. 

* Meaning in the present connexion momentum of translatory 
motion or linear momentum, as distinguished from the angular 
momentum of rotatory motion. Cf. Art. 69. The law holds in an 
extended sense for both together. Cf. Art. 70. 



iv] MASS-VECTORS 49 

67. BY THE INTRODUCTION OF THE IDEA OF MASS WE 
PASS FROM POINT- VECTORS, POINT DISPLACEMENTS, 
VELOCITIES, TOTAL ACCELERATIONS, AND RATES OF 
ACCELERATION, TO MASS-VECTORS, MASS DISPLACE- 
MENTS, MOMENTA, IMPULSES, AND MOVING FORCES. 
In the diagram -of rates _of acceleration (Fig. 9, 
Article 63) the vectors coa, aj/3, etc., drawn from the 
origin, represent the rates of acceleration of the bodies 
A, B, etc., at a given instant, with respect to that of 
the origin O. 

The corresponding mass-vectors, cua . A, a>jS . B, etc., 
represent the forces acting on the bodies A, B, etc. 

We sometimes speak of several forces acting on a body, 
when the force acting on the body arises from several 
different causes, so that we naturally consider the parts 
of the force arising from these different causes separately. 
But when we consider force, not with respect to its 
causes, but with respect to its effect that of altering 
the motion of a body we speak not of the forces, but 
of the force acting on the body, and this force is 
measured by the rate of change of the momentum of 
the body, and is indicated by the mass-vector in the 
diagram of rates of acceleration*. 

* This distinction is conveniently expressed by the terms 
applied forces and effective forces. For a single particle these two 
sets are statically equivalent. Therefore for any body which can 
be regarded as a system of particles held together by mutual 
influences, the same must be true in the aggregate, when their 
mutual forces are also included among the applied forces. But these 
internal mutual forces must in any case immediately become 
adjusted so as to be statically equilibrated by themselves, other- 
wise the parts of the body would be set by them into continually 
accelerated motion even when it is removed from all external 
influences. Therefore, leaving them out of account, the forces 
applied from without are statically equivalent, as regards the 
given type of body, to the effective forces that accelerate the 
particles or elements of mass of that body. This is the Principle 
of d'Alembert: though it is implied in the Newtonian scheme, 
being provided for by the Third Law, its more explicit recognition 
in 1743 gave rise to great simplification in the treatment of 
abstruse dynamical problems, as exemplified in d'Alembert's 
discussion of the spin of the earth's axis which causes the pre- 
cession of the equinoxes, by reducing them to problems of statics. 
M. 4 



50 ON CENTRE OF MASS [CH. 

We have thus a series of different kinds of mass- 
vectors corresponding to the series of vectors which we 
have already discussed. 

We have, in the first place, a system of mass- vectors 
with a common origin, which we may regard as a 
method of indicating the distribution of mass in a 
material system, just as the corresponding system of 
vectors indicates the geometrical configuration of the 
system. 

In the next place, by comparing the distribution of 
mass at two different epochs, we obtain a system of 
mass-vectors of displacement. 

The rate of mass displacement is momentum, just as 
the rate of displacement is velocity. 

The change of momentum is impulse, as the change 
of velocity is total acceleration. 

The rate of change of momentum is moving force, as 
the rate of change of velocity is rate of acceleration. 

68. DEFINITION OF A MASS- AREA 
When a material particle moves from one point to 
another, twice the area swept out by the vector of the 
particle multiplied by the mass of the particle is callecl 
the mass-area of the displacement of the particle with 
respect to the origin from which the vector is drawn. 

If the area is in one plane, the direction of the mass- 
area is normal to the plane, drawn so that, looking in 
the positive direction along the normal, the motion of 
the particle round its area appears to be the direction 
of the motion of the hands of a watch*. 

If the area is not in one plane, the path of the 
particle must be divided into portions so small that 
each coincides sensibly with a straight line, and the 
mass-areas corresponding to these portions must be 
added together by the rule for the addition of vectors. 

* Stated in absolute terms, the motion round the area is in 
the direction of a right-handed screw motion which progresses 
along the normal in the positive direction. 



iv] ANGULAR MOMENTUM 51 

69. ANGULAR MOMENTUM 

The rate of change of a mass-area is twice the mass 
of the particle into the triangle, whose vertex is the 
origin and whose base is the velocity of the particle 
measured along the line through the particle in the 
direction of its motion. The direction of this mass- 
area is indicated by the normal drawn according to the 
rule given above. 

The rate of change of the mass-area of a particle is 
called the Angular Momentum of the particle about the 
origin, and the sum of the angular momenta of all the 
particles is called the angular momentum of the system 
about the origin. 

The angular momentum of a material system with 
respect to a point is, therefore, a quantity having a 
definite direction as well as a definite magnitude. 

The definition of the angular momentum of a particle 
about a point may be expressed somewhat differently 
as the product of the momentum of the particle with 
respect to that point into the perpendicular from that 
point on the line of motion of the particle at that 
instant. 

70. MOMENT OF A FORCE ABOUT A POINT 
The rate of increase of the angular momentum of 
a particle is the continued product of the rate of 
acceleration of the velocity of the particle into the 
mass of the particle into the perpendicular from the 
origin on the line through the particle along which 
the acceleration takes place. In other words, it is the 
product of the moving force acting on the particle into 
the perpendicular from the origin on the line of action 
of this force. 

Now the product of a force into the perpendicular 
from the origin on its line of action is called the 
Moment of the force about the origin. The axis of the 
moment, which indicates its direction, is a vector 
drawn perpendicular to the plane passing through the 

42 



52 ON CENTRE OF MASS [CH. 

force and the origin, and in such a direction that, 
looking along this line in the direction in which it is 
drawn, the force tends to move the particle round the 
origin in the direction of the hands of a watch. 

Hence the rate of change of the angular momentum 
of a particle about the origin is measured by the 
moment of the force which acts on the particle about 
that point. 

The rate of change of the angular momentum of a 
material system about the origin is in like manner 
measured by the geometric sum of the moments 
of the forces which act on the particles of the system. 

71. CONSERVATION OF ANGULAR MOMENTUM 

Now consider any two particles of the system. The 
forces acting on these two particles, arising from their 
mutual action, are equal, opposite, and in the same 
straight line. Hence the moments of these forces about 
any point as origin are equal, opposite, and about the 
same axis. The sum of these moments is therefore zero. 
In like manner the mutual action between every 
other pair of particles in the system consists of two 
forces, the sum of whose moments is zero. 

Hence the mutual action between the bodies of a 
material system does not affect the geometric sum of 
the moments of the forces. The only forces, therefore, 
which need be considered in finding the geometric sum 
of the moments are those which are external to the 
system that is to say, between the whole or any 
part of the system and bodies not included in the 
system. 

The rate of change of the angular momentum of the 
system is therefore measured by the geometric sum of 
the moments of the external forces acting on the 
system. 

If the directions of all the external forces pass through 
the origin, their moments are zero, and the angular 
momentum of the system will remain constant. 



iv] ANGULAR MOMENTUM 53 

When a planet describes an orbit about the sun, 
the direction of the mutual action between the two 
bodies always passes through their common centre of 
mass. Hence the angular momentum of either body 
about their common centre of mass remains constant, 
so far as these two bodies only are concerned, though 
it may be affected by the action of other planets. If, 
however, we include all the planets in the system, the 
geometric sum of their angular momenta about their 
common centre of mass will remain absolutely con- 
stant*, whatever may be their mutual actions, provided 
no force arising from bodies external to the whole solar 
system acts in an unequal manner upon the different 
members of the system. 

* That is, the plane of the total angular momentum of the 
solar system is invariable in direction in space. 

The plane of this resultant angular momentum, called by 
Laplace the "invariable plane," is fundamental for the exact 
specification of the motion of the solar system. 



CHAPTER V 
ON WORK AND ENERGY 

72. DEFINITIONS 

WORK is the act of* producing a change of configuration 
in a system in opposition to a force which resists that 
change. 

ENERGY is the capacity of doing work. 

When the nature of a material system is such that 
if, after the system has undergone any series of changes 
it is brought back in any manner to its original state, 
the whole work done by external agents on the system 
is equal to the whole work done by the system in over- 
coming external forces, the system is called a CON- 
SERVATIVE SYSTEMf . 

73. PRINCIPLE OF CONSERVATION OF ENERGY 

The progress of physical science has led to the dis- 
covery and investigation of different forms of energy, 
and to the establishment of the doctrine that all 
material systems may be regarded as conservative 
systems, provided that all the different forms of energy 
which exist in these systems are taken into account. 

This doctrine, considered as a deduction from ob- 
servation and experiment, can, of course, assert no 
more than that no instance of a non-conservative 
system has hitherto been discovered. 

As a scientific or science-producing doctrine, how- 

* The work done is a quantitative measure of the effort ex- 
pended in deranging the system, in terms of the consumption of 
energy that is required to give effect to it. 

The idea of work implies a fund of energy, from which the work 
is supplied. 

f As distinguished from a system in which the energy available 
for work becomes gradually degraded to less available forms by 
frictional agencies, called a Dissipative System. Cf. Art. 93. 



CH.V] CONSERVATION 55 

ever, it is always acquiring additional credibility from 
the constantly increasing number of deductions which 
have been drawn from it, and which are found in all 
cases to be verified by experiment. 

In fact the doctrine of the Conservation of Energy is 
the one generalised statement which is found to be 
consistent with fact, not in one physical science only, 
but in all. 

When once apprehended it furnishes to the physical 
inquirer a principle on which he may hang every known 
law relating to physical actions, and by which he may 
be put in the way to discover the relations of such 
actions in new branches of science*. 

For such reasons the doctrine is commonly called the 
Principle of the Conservation of Energy. 

74. GENERAL STATEMENT OF THE PRINCIPLE OF 
THE CONSERVATION OF ENERGY 

The total energy of any material system is a quantity 
which can neither be increased nor diminished by any 
action between the -parts of the system, though it may be 
transformed into any of the forms of which energy is 
susceptible. 

If, by the action of some agent external to the 
system, the configuration of the system is changed, 
while the forces of the system resist this change of 
configuration, the external agent is said to do work on 
the system. In this case the energy of the system is 
increased by the amount of work done on it by the 
external agent. 

If, on the contrary, the forces of the system produce 
a change of configuration which is resisted by the 
external agent, the system is said to do work on the 

* Every law relating to the forces of statical or steady systems 
is involved implicitly in the complete expression for the Energy 
of the system. But in a kinetic system, where force is being used 
in producing energy of motion, a more elaborate principle is re- 
quired, that of Least Action, for example. See infra, Chapter ix. 



56 ON ENERGY [CH. 

external agent, and the energy of the system is dimin- 
ished by the amount of work which it does. 

Work, therefore, is a transference of energy from 
one system to another; the system which gives out 
energy is said to do work on the system which receives 
it, and the amount of energy given out by the first 
system is always exactly equal to that received by the 
second. 

If, therefore, we include both systems in one larger 
system, the energy of the total system is neither 
increased nor diminished by the action of the one 
partial system on the other. 

75. MEASUREMENT OF WORK 

Work done by an external agent on a material system 
may be described as a change* in the configuration of 
the system taking place under the action of an external 
force tending to produce that change. 

Thus, if one pound is lifted one foot from the ground 
by a man in opposition to the force of gravity, a certain 
amount of work is done by the man, and this quantity 
is known among engineers as one foot-pound. 

Here the man is the external agent, the material 
system consists of the earth and the pound, the change of 
configuration is the increase of the distance between 
the matter of the earth and the matter of the pound, 
and the force is the upward force exerted by the man in 
lifting the pound, which is equal and opposite to the 
weight of the pound. To raise the pound a foot higher 
would, if gravity were a uniform force, require exactly 
the same amount of work. It is true that gravity is not 
really uniform, but diminishes as we ascend from the 
earth's surface, so that a foot-pound is not an accurately 

* See footnote, Art. 72. 

These ideas, leading to an estimate of the total effect by work 
done rather than momentum , produced, are of the kind that 
were enforced by Leibniz. What was then mainly needed to avoid 
confusion was a set of names for the different effects. 



v] WORK 57 

known quantity, unless we specify the intensity of 
gravity at the place. But for the purpose of illustration 
we may assume that gravity is uniform for a few feet of 
ascent, and in that case the work done in lifting a pound 
would be one foot-pound for every foot the pound is 
lifted. 

To raise twenty pounds of water ten feet high 
requires 200 foot-pounds of work. To raise one pound 
ten feet high requires ten foot-pounds, and as there are 
twenty pounds the whole work is twenty times as 
much, or two hundred foot-pounds. 

The quantity of work done is, therefore, proportional 
to the product of the numbers representing the force 
exerted and the displacement in the direction of the 
force. 

In the case of a foot-pound the force is the weight of 
a pound a quantity which, as we know, is different in 
different places. The weight of a pound expressed in 
absolute measure is numerically equal to the intensity 
of gravity, 'the quantity denoted by g, the value of 
which in poundals to the pound varies from 32-227 at 
the poles to 32-117 at the equator, and diminishes 
without limit as we recede from the earth. In dynes 
to the gramme it varies from 978-1 to 983-1. Hence, 
in order to express work in a uniform and consistent 
manner, we must multiply the number of foot-pounds 
by the number representing the intensity of gravity at 
the place. The work is thus reduced to foot-poundals. 
We shall always understand work to be measured in 
this manner and reckoned in foot-poundals when no 
other system of measurement is mentioned. When 
work is expressed in foot-pounds the system is that of 
gravitation-measures y which is not a complete system 
unless we also know the intensity of gravity at the 
place. 

In the metrical system the unit of work is the Erg, 
which is the work done by a dyne acting through a 
centimetre. There are 421393-8 ergs in a foot-poundal. 



58 ON ENERGY [CH. 

76. POTENTIAL ENERGY 

The work done by a man in raising a heavy body is 
done in overcoming the attraction between the earth 
and that body. The energy of the material system, 
consisting of the earth and the heavy body, is thereby 
increased. If the heavy body is the leaden weight of a 
clock, the energy of the clock is increased by winding 
it up, so that the clock is able to go for a week in spite 
of the friction of the wheels and the resistance of the 
air to the motion of the pendulum, and also to give out 
energy in other forms, such as the communication of 
the vibrations to the air, by which we hear the ticking 
of the clock. 

When a man winds up a watch he does work in 
changing the form of the mainspring by coiling it up. 
The energy of the mainspring is thereby increased, so 
that as it uncoils itself it is able to keep the watch 
going. 

In both these cases the energy communicated to the 
system depends upon a change of configuration. 

77. KINETIC ENERGY 

But in a very important class of phenomena the work 
is done in changing the velocity of the body on which it 
acts. Let us take as a simple case that of a body 
moving without rotation under the action of a force. 
Let the mass of the body be M pounds, and let a force 
of F poundals act on it in the line of motion during an 
interval of time, T seconds. Let the velocity at the 
beginning of the interval be V and that at the end V 
feet per second, and let the distance travelled by the 
body during the time be S feet. The original momen- 
tum is MV, and the final momentum is MV ', so that 
the increase of momentum is M (V F), and this, by 
the second law of motion, is equal to FT, the impulse 
of the force F acting for the time T. Hence 

FT=M(V - V] (i). 



v] KINETIC ENERGY 59 

Since the velocity increases uniformly with the time 
[when the force is constant], the mean velocity is the 
arithmetical mean of the original and final velocities, 
or^(V+ V). 

We can also determine the mean velocity by dividing 
the space S by the time T, during which it is described. 
Hence 

T=l(V+V) (2). 

Multiplying the corresponding members of equations 
(i) and (2) each by each we obtain 

FS= IMV'i-lMV 2 (3). 

Here FS is the work done by the force F acting on the 
body while it moves through the space S in the direction 
of the force, and this is equal to the excess of %MV' 2 
above %MV Z . If we call \M V 2 , or half the product of 
the mass into the square of the velocity, the kinetic 
energy of the body at first, then \MV 2 will be the 
kinetic energy after the action of the force F through 
the space S. The energy is here expressed in foot- 
poundals. 

We may now express the equation in words by 
saying that the work done by the force F in changing 
the motion of the body is measured by the increase of 
the kinetic energy of the body during the time that the 
force acts. 

We have proved that this is true, when the interval of 
time is so small that we may consider the force as 
constant during that time, and the mean velocity during 
the interval as the arithmetical mean of the velocities at 
the beginning and end of the interval. This assumption, 
which is exactly true when the force is constant, how- 
ever long the interval may be, becomes in every case 
more and more nearly true as the interval of time taken 
becomes smaller and smaller. By dividing the whole 
time of action into small parts, and proving that in 
each of these the work done is equal to the increase of 



60 ON ENERGY [CH. 

the kinetic energy of the body, we may, by adding the 
successive portions of the work and the successive 
increments of energy, arrive at the result that the total 
work done by the force is equal to the total increase of 
kinetic energy. 

If the force acts on the body in the direction opposite 
to its motion, the kinetic energy of the body will be 
diminished instead of being increased, and the force, 
instead of doing work on the body, will act as a resist- 
ance, which the body, in its motion, overcomes. Hence 
a moving body, as long as it is in motion, can do work in 
overcoming resistance, and the work done by the moving 
body is equal to the diminution of its kinetic energy, 
till at last, when the body is brought to rest, its kinetic 
energy is exhausted, and the whole work it has done 
is then equal to the whole kinetic energy which it had 
at first. 

We now see the appropriateness of the name kinetic 
energy, which we have hitherto used merely as a name 
to denote the product |MF 2 . For the energy of a body 
has been defined as the capacity which it has of doing 
work, and it is measured by the work which it can do. 
The kinetic energy of a body is the energy it has in 
virtue of being in motion, and we have now shown that 
its value is expressed by \MV Z or \M V x V, that is, 
half the product of its momentum into its velocity. 

78. OBLIQUE FORCES 

If the force acts on the body at right angles to the 
direction of its motion it does no work on the body, and 
it alters the direction but not the magnitude of the 
velocity. The kinetic energy, therefore, which depends 
on the square of the velocity, remains unchanged. 

If the direction of the force is neither coincident with, 
nor at right angles to, that of the motion of the body we 
may resolve the force into two components, one of which 
is at right angles to the direction of motion, while the 



v] OF A BODY IN MOTION 61 

other is in the direction of motion (or in the opposite 
direction). 

The first of these components may be left out of 
consideration in all calculations about energy, since it 
neither does work on the body nor alters its kinetic 
energy. 

The second component is that which we have already 
considered. When it is in the direction of motion it 
increases the kinetic energy of the body by the amount 
of work which it does on the body. When it is in the 
opposite direction the kinetic energy of the body is 
diminished by the amount of work which the body does 
against the force. 

Hence in all cases the increase of kinetic energy is 
equal to the work done on the body by external agency, 
and the diminution of kinetic energy is equal to the 
work done by the body against external resistance. 



79. KINETIC ENERGY OF Two PARTICLES REFERRED 
TO THEIR CENTRE OF MASS 

The kinetic energy of a material system is equal to 
the kinetic energy of a mass equal to that of the system 
moving with the velocity of the centre of mass of the 
system, together with the kinetic energy due to the 
motion of the parts of the system , 

relative to its centre of mass. 

Let us begin with the case of 
two particles whose masses are A 
and B, and whose velocities are 
represented in the diagram of 
velocities by the lines oa and ob. 
If c is the centre of mass of a Fig. 10. 

particle equal to A placed at a, 
and a particle equal to B placed at 6, then oc will 
represent the velocity of the centre of mass of the two 
particles. 




62 ON ENERGY [CH. 

The kinetic energy of the system is the sum of the 
kinetic energies of the particles, or 

T = lAoa 2 - + %Bob 2 . 

Expressing oa 2 and ob 2 in terms of oc, ca and cb, and 
the angle oca = 9, 

T = Aoc 2 + lAca? - Aoc.ca cos 
+ %Boc 2 + IBcb 2 - Boc.cb cos d. 
But since c is the centre of mass of A at a, and B at b, 

Aca + Bcb = o. 
Hence adding 

T = i (A + B) oc 2 + iAca 2 + Bcb 2 , 
or, the kinetic energy of the system of two particles A 
and B is equal to that of a mass equal to (A + B) 
moving with the velocity of the centre of mass, together 
with that of the motion of the particles relative to the 
centre of mass. 

80. KINETIC ENERGY OF A MATERIAL SYSTEM 

REFERRED TO ITS CENTRE OF MASS 
We have begun with the case of two particles, because 
the motion of a particle is assumed to be that of its 
centre of mass, and we have proved our proposition 
true for a system of two particles. But if the proposi- 
tion is true for each of two material systems taken 
separately, it must be true of the system which they 
form together. For if we now suppose oa and ob to 
represent the velocities of the centres of mass of two 
material systems A and B, then oc will represent the 
velocity of the centre of mass of the combined system 
A + B, and if T A represents the kinetic energy of the 
motion of the system A relative to its own centre of 
mass, and TB the same for the system B, then if the 
proposition is true for the systems A and B taken 
separately, the kinetic energy of A is 

Aoa 2 + T A , 
and that of B Bob 2 + T B . 



v] OF A MATERIAL SYSTEM 63 

The kinetic energy of the whole is therefore 

T B 



The first term represents the kinetic energy of a mass 
equal to that of the whole system moving with the 
velocity of the centre of mass of the whole system. 

The second and third terms, taken together, represent 
the kinetic energy of the system A relative to the centre 
of gravity of the whole system, and the fourth and 
fifth terms represent the same for the system B. 

Hence if the proposition is true for the two systems 
A and B taken separately, it is true for the system 
compounded of A and B. But we have proved it true 
for the case of two particles; it is therefore true for 
three, four, or any other number of particles, and there- 
fore for any material system. 

The kinetic energy of a system referred to its centre 
of mass is less than its kinetic energy when referred to 
any other point. 

For the latter quantity exceeds the former by a 
quantity equal to the kinetic energy of a mass equal to 
that of the whole system moving with the velocity of 
the centre of mass relative to the other point, and since 
all kinetic energy is essentially positive, this excess must 
be positive. 

81. AVAILABLE KINETIC ENERGY 

We have already seen in Article 64 that the mutual 
action between the parts of a material system cannot 
change the velocity of the centre of mass of the system. 
Hence that part of the kinetic energy of the system 
which depends on the motion of the centre of mass 
cannot be affected by any action internal to the system. 
It is therefore impossible, by means of the mutual 
action of the parts of the system, to convert this part 
of the energy into work. As far as the system itself 
is concerned, this energy is unavailable. It can be 



64 ON ENERGY [CH. 

converted into work only by means of the action 
between this system and some other material system 
external to it. 

Hence if we consider a material system unconnected 
with any other system, its available kinetic energy is 
that which is due to the motions of the parts of the 
system relative to its centre of mass. 

Let us suppose that the action between the parts of 
the system is such that after a certain time the con- 
figuration of the system becomes invariable, and let us 
call this process the solidification of the system. We 
have shown that the angular momentum of the whole 
system is not changed by any mutual action of its parts. 
Hence if the original angular momentum is zero, the 
system, when its form becomes invariable, will not rotate 
about its centre of mass, but if it moves at all will move 
parallel to itself, and the parts will be at rest relative 
to the centre of mass. In this case therefore the whole 
available energy will be converted into work by the 
mutual action of the parts during the solidification of 
the system. 

If the system has angular momentum, it will have 
the same angular momentum when solidified. It will 
therefore rotate about its centre of mass, and will 
therefore still have energy of motion relative to its 
centre of mass, and this remaining kinetic energy has 
not been converted into work. 

But if the parts of the system are allowed to separate 
from one another in directions perpendicular to the 
axis of the angular momentum of the system, and if the 
system when thus expanded is solidified, the remaining 
kinetic energy of rotation round the centre of mass 
will be less and less the greater the expansion of the 
system, so that by sufficiently expanding the system 
[before it is solidified] we may make the remaining 
kinetic energy as small as we please, so that the whole 
kinetic energy relative to the centre of mass of the 
system may be converted into work within the system. 



v] POTENTIAL ENERGY 65 

82. POTENTIAL ENERGY 

The potential energy of a material system is the 
capacity which it has of doing work [on other systems] 
depending on other circumstances than the motion 
of the system. In other words, potential energy is that 
energy which is not kinetic. 

In the theoretical material system which we build up 
in our imagination from the fundamental ideas of matter 
and motion, there are no other conditions present except 
the configuration and motion of the different masses of 
which the system is composed. Hence in such a system 
the circumstances upon which the energy must depend 
are motion and configuration only, so that, as the kinetic 
energy depends on the motion, the potential energy 
must depend on the configuration. 

In many real material systems we know that part of 
the energy does depend on the configuration. Thus 
the mainspring of a watch has more energy when 
coiled up than when partially uncoiled, and two bar 
magnets have more energy when placed side by side 
with their similar poles turned the same way than when 
their dissimilar poles are placed next each other. 

83. ELASTICITY 

In the case of the spring we may trace the connexion 
between the coiling of the spring and the force which 
it exerts somewhat further by conceiving the spring 
divided (in imagination) into very small parts or ele- 
ments. When the spring is coiled up, the form of each 
of these small parts is altered, and such an alteration of 
the form of a solid body is called a Strain. 

In solid bodies strain is accompanied with internal 
force or stress ; those bodies in which the stress depends 
simply on the strain are called Elastic, and the property 
of exerting stress when strained is called Elasticity. 

We thus find that the coiling of the spring involves 
the strain of its elements, and that the external force 



66 ON ENERGY [CH. 

which the spring exerts is the resultant of the stresses 
in its elements. 

We thus substitute for the immediate relation 
between the coiling of the spring and the force which it 
exerts, a relation between the strains and stresses of 
the elements of the spring ; that is to say, for a single 
displacement and a single force, the relation between 
which may in some cases be of an exceedingly compli- 
cated nature, we substitute a multitude of strains and 
an equal number of stresses, each strain being con- 
nected with its corresponding stress by a much more 
simple relation. 

But when all is done, the nature of the connexion 
between configuration and force remains as mysterious 
as ever. We can only admit the fact, and if we call 
all such phenomena phenomena of elasticity, we may 
find it very convenient to classify them in this way, 
provided we remember that by the use of the word 
elasticity we do not profess to explain the cause of the 
connexion between configuration and energy. 



84. ACTION AT A DISTANCE 

In the case of the two magnets there is no visible 
substance connecting the bodies between which the 
stress exists. The space between the magnets may be 
filled with air or with water, or we may place the magnets 
in a vessel and remove the air by an air-pump, till the 
magnets are left in what is commonly called a vacuum, 
and yet the mutual action of the magnets will not be 
altered. We may even place a solid plate of glass or 
metal or wood between the magnets, and still we find 
that their mutual action depends simply on their 
relative position, and is not perceptibly modified by 
placing any substance between them, unless that 
substance is one of the magnetic metals. Hence the 
action between the magnets is commonly spoken of as 
action at a distance. 



v] POTENTIAL AND KINETIC 67 

Attempts have been made, with a certain amount of 
success 1 , to analyse this action at a distance into a 
continuous distribution of stress in an invisible medium, 
and thus to establish an analogy between the magnetic 
action and the action of a spring or a rope in trans- 
mitting force; but still the general fact that strains or 
changes of configuration are accompanied by stresses or 
internal forces, and that thereby energy is stored up 
in the system so strained, remains an ultimate fact 
which has not yet been explained as the result of any 
more fundamental principle. 

85. THEORY OF POTENTIAL ENERGY MORE COMPLI- 
CATED THAN THAT OF KlNETIC ENERGY 

Admitting that the energy of a material system may 
depend on its configuration, the mode in which it so 
depends may be much more complicated than the mode 
in which the kinetic energy depends on the motion of 
the system. For the kinetic energy may be calculated 
from the motion of the parts of the system by an in- 
variable method. We multiply the mass of each part by 
half the square of its velocity, and take the sum of all 
such products. But the potential energy arising from 
the mutual action of two parts of the system may 
depend on the relative position of the parts in a manner 
which may be different in different instances. Thus 
when two billiard balls approach each other from a 
distance, there is no sensible action between them till 
they come so near one another that certain parts appear 
to be in contact. To bring the centres of the two balls 
nearer, the parts in contact must be made to yield, and 
this requires the expenditure of work. 

1 See Clerk Maxwell's Treatise on Electricity and Magnetism, 
Vol. II, Art. 641. [Modern scrutiny requires a distribution of 
momentum in the medium, which reveals itself for example in 
the pressure of radiation, in addition to the stress: cf. appendix to 
J. H. Poynting's Collected Papers. It in fact develops into the 
guiding tensor principle in the theory of gravitational relativity.] 

52 



68 ON ENERGY [CH. 

Hence in this case the potential energy is constant 
for all distances greater than the distance of first 
contact, and then rapidly increases when the distance 
is diminished. 

The force between magnets varies with the distance 
in a very different manner, and in fact we find that it is 
only by experiment that we can ascertain the form of 
the relation between the configuration of a system and 
its potential energy. 

86. APPLICATION OF THE METHOD OF ENERGY TO 
THE CALCULATION OF FORCES 

A complete knowledge of the mode in which the 
energy of a material system varies when the configura- 
tion and motion of the system are made to vary is 
mathematically equivalent to a knowledge of all the 
dynamical properties of the system. The mathematical 
methods by which all the forces and stresses in a moving 
system are deduced from the single mathematical 
formula which expresses the energy as a function of the 
variables have been developed by Lagrange, Hamilton, 
and other eminent mathematicians, but it would be 
difficult even to describe them in terms of the elementary 
ideas to which we restrict ourselves in this book. An 
outline of these methods is given in my treatise on 
Electricity, Part IV, Chapter V, Article 533*, and the 
application of these dynamical methods to electro- 
magnetic phenomena is given in the chapters im- 
mediately following. 

But if we consider only the case of a system at rest 
it is easy to see how we can ascertain the forces of the 
system when we know how its energy depends on its 
configuration. 

For let us suppose that an agent external to the 
system produces a displacement from one configuration 
to another, then if in the new configuration the system 

* Reprinted infra, p. 123. 



v] FORCE IN TERMS OF ENERGY 69 

possesses more energy than it did at first, it can have 
received this increase of energy only from the external 
agent. This agent must therefore have done an amount 
of work equal to the increase of energy. It must 
therefore have exerted force in the direction of the 
displacement, and the mean value of this force, multi- 
plied into the displacement, must be equal to the work 
done. Hence the mean value of the force may be found 
by dividing the increase of energy by the displacement. 

If the displacement is large this force may vary con- 
siderably during the displacement, so that it may be 
difficult to calculate its mean value ; but since the force 
depends on the configuration, if we make the displace- 
ment smaller and smaller the variation of the force will 
become smaller and smaller, so that at last the force 
may be regarded as sensibly constant during the dis- 
placement. 

If, therefore, we calculate for a given configuration 
the rate at which the energy increases with the dis- 
placement, by a method similar to that described in 
Articles 27, 28, and 33, this rate will be numerically 
equal to the force exerted by the external agent in the 
direction of the displacement. 

If the energy diminishes instead of increasing as the 
displacement increases, the system must do work on 
the external agent, and the force exerted by the external 
agent must be in the direction opposite to that of dis- 
placement. 

87. SPECIFICATION OF THE [MODE OF ACTION] 

OF FORCES 

In treatises on dynamics the forces spoken of are 
usually those exerted by the external agent on the 
material system. In treatises on electricity, on the 
other hand, the forces spoken of are usually those 
exerted by the electrified system against an external 
agent which prevents the system from moving. It is 
necessary, therefore, in reading any statement about 



70 ON ENERGY [CH. 

forces, to ascertain whether the force spoken of is to 
be regarded from the one point of view or the other. 

We may in general avoid any ambiguity by viewing 
the phenomenon as a whole, and speaking of it as a 
stress exerted between two points or bodies, and dis- 
tinguishing it as a tension or a pressure, an attraction or 
a repulsion, according to its direction. See Article 55. 

88. APPLICATION TO A SYSTEM IN MOTION 

It thus appears that from a knowledge of the potential 
energy of a system in every possible configuration 
we may deduce all the external forces which are re- 
quired to keep the system in [any given] configuration. 
If the system is at rest, and if these external forces are 
the actual forces, the system will remain in equilibrium. 
If the system is in motion the force acting on each 
particle is that arising from the connexions of the 
system (equal and opposite to the external force just 
calculated), together with any external force which may 
be applied to it. Hence a complete knowledge of the 
mode in which the potential energy varies with the 
configuration would enable us to predict every possible 
motion of the system under the action of given external 
forces, provided we were able to overcome the purely 
mathematical difficulties of the calculation. 

89. APPLICATION OF THE METHOD OF ENERGY TO 
THE INVESTIGATION OF REAL BODIES 

When we pass from abstract dynamics to physics 
from material systems, whose only properties are those 
expressed by their definitions, to real bodies, whose 
properties we have to investigate we find that there 
are many phenomena which we are not able to explain 
as changes in the configuration and motion of a material 
system. 

Of course if we begin by assuming that the real 
bodies are systems composed of matter which agrees 
in all respects with the definitions we have laid down, 



v] EXPERIMENTAL METHOD 71 

we may go on to assert that all phenomena are changes 
of configuration and motion, though we are not pre- 
pared to define the kind of configuration and motion by 
which the particular phenomena are to be explained. 
But in accurate science such asserted explanations must 
be estimated, not by their promises, but by their per- 
formances. The configuration and motion of a system 
are facts capable of being described in an accurate 
manner, and therefore, in order that the explanation of 
a phenomenon by the configuration and motion of a 
material system may be admitted as an addition to our 
scientific knowledge, the configurations, motions, and 
forces must be specified, and shown to be consistent 
with known facts, as well as capable of accounting for 
the phenomenon. 

90. VARIABLES ON WHICH THE ENERGY DEPENDS 

But even when the phenomena we are studying 
have not yet been explained dynamically, we are still 
able to make great use of the principle of the conserva- 
tion of energy as a guide to our researches. 

To apply this principle, we in the first place assume 
that the quantity of energy in a material system depends 
on the state of that system, so that for a given state 
there is a definite amount of energy. 

Hence the first step is to define the different states 
of the system, and when we have to deal with real 
bodies we must define their state with respect not only 
to the configuration and motion of their visible parts, 
but if we have reason to suspect that the configuration 
and motion of their invisible particles influence the 
visible phenomenon, we must devise some method of 
estimating the energy thence arising. 

Thus pressure, temperature, electric potential, and 
chemical composition are variable quantities, the values 
of which serve to specify the state of a body, and in 
general the energy of the body depends on the values 
of these and other variables. 



72 ON ENERGY [CH. 

91. ENERGY IN TERMS OF THE VARIABLES 

The next step in our investigation is to determine 
how much work must be done by external agency on 
the body in order to make it pass from one specified 
state to another. 

For this purpose it is sufficient to know the work 
required to make the body pass from a particular state, 
which we may call the standard state, into any other 
specified state. The energy in the latter state is equal 
to that in the standard state, together with the work 
required to bring it from the standard state into the 
specified state. The fact that this work is the same 
through whatever series of states the system has passed 
from the standard state to the specified state is the 
foundation of the whole theory of energy. 

Since all the phenomena depend on the variations of 
the energy of the body, and not on its total value, it is 
unnecessary, even if it were possible, to form any 
estimate of the energy of the body in its standard state. 

92. THEORY OF HEAT 

One of the most important applications of the prin- 
ciple of the conservation of energy is to the investigation 
of the nature of heat. 

At one time it was supposed that the difference be- 
tween the states of a body when hot and when cold was 
due to the presence of a substance called caloric, which 
existed in greater abundance in the body when hot than 
when cold. But the experiments of Rumford on the 
heat produced by the friction of metal, and of Davy on 
the melting of ice by friction, have shown that when 
work is spent in overcoming friction, the amount of heat 
produced is proportional to the work spent. 

The experiments of Hirn have also shown that when 
heat is made to do work in a steam-engine, part of the 
heat disappears, and that the heat which disappears is 
proportional to the work done. 



v] HEAT 73 

A very careful measurement of the work spent in 
friction, and of the heat produced, has been made by 
Joule, who finds that the heat required to raise one 
pound of water from 39 F. to 40 F. is equivalent to 
772 foot-pounds of work at Manchester, or 24,858 foot- 
poundals. 

From this we may find that the heat required to 
raise one gramme of water from 3 C. to 4 C. is 
42,000,000 ergs. 

93 . HEAT A FORM OF ENERGY 

Now, since heat can be produced it cannot be a sub- 
stance ; and since whenever mechanical energy is lost by 
friction there is a production of heat, and whenever 
there is a gain of mechanical energy in an engine there 
is a loss of heat ; and since the quantity of energy lost 
or gained is proportional to the quantity of heat gained 
or lost, we conclude that heat is a form of energy. 

We have also reasons for believing that the minute 
particles of a hot body are in a state of rapid agitation, 
that is to say, that each particle is always moving very 
swiftly, but that the direction of its motion alters so 
often that it makes little or no progress from one region 
to another. 

If this be the case, a part, and it may be a very large 
part, of the energy of a hot body must*be in the form 
of kinetic energy. 

But for our present purpose it is unnecessary to 
ascertain in what form energy exists in a hot body ; the 
most important fact is that energy may be measured in 
the form of heat, and since every kind of energy may 
be converted into heat, this gives us one of the most 
convenient methods of measuring it. 

94. ENERGY MEASURED AS HEAT 

Thus when certain substances are placed in contact 
chemical actions take place, the substances combine in 
a new way, and the new group of substances has differ- 



74 ON ENERGY [CH. 

ent chemical properties from the original group of 
substances. During this process mechanical work may 
be done by the expansion of the mixture, as when 
gunpowder is fired ; an electric current may be produced, 
as in the voltaic battery; and heat may be generated, 
as in most chemical actions. 

The energy given out in the form of mechanical 
work may be measured directly, or it may be trans- 
formed into heat by friction. The energy spent in 
producing the electric current may be estimated as 
heat by causing the current to flow through a conductor 
of such a form that the heat generated in it can easily 
be measured. Care must be taken that no energy is 
transmitted to a distance in the form of sound or 
radiant heat without being duly accounted for. 

The energy remaining in the mixture, together with 
the energy which has escaped, must be equal to the 
original energy. 

Andrews, Favre and Silbermann, [Julius Thomsen,] 
and others, have measured the quantity of heat pro- 
duced when a certain quantity of oxygen or of chlorine 
combines with its equivalent of other substances. These 
measurements enable us to calculate the excess of the 
energy which the substances concerned had in their 
original state, when uncombined, above that which they 
have after combination. 

95. SCIENTIFIC WORK TO BE DONE 

Though a great deal of excellent work of this kind 
has already been done, the extent of the field hitherto 
investigated appears quite insignificant when we con- 
sider the boundless variety and complexity of the 
natural bodies with which we have to deal. 

In fact the special work which lies before the physical 
inquirer in the present state of science is the deter- 
mination of the quantity of energy which enters or 
leaves a material system during the passage of the sys- 
tem from its standard state to any other definite state. 



v] HISTORY 75 

96. HISTORY OF THE DOCTRINE OF ENERGY 

The scientific importance of giving a name to the 
quantity which we call kinetic energy seems to have 
been first recognised by Leibniz, who gave to the 
product of the mass by the square of the velocity the 
name of Vis Viva. This is twice the kinetic energy. 

Newton, in the " Scholium to the Laws of Motion," 
expresses the relation between the rate at which work 
is done by the external agent, and the rate at which 
it is given out, stored up, or transformed by any machine 
or other material system, in the following statement, 
which he makes in order to show the wide extent of the 
application of the Third Law of Motion. 

" If the action of the external agent is estimated by 
the product of its force into its velocity, and the re- 
action of the resistance in the same way by the product 
of the velocity of each part of the system into the 
resisting force arising from friction, cohesion, weight, 
and acceleration, the action and reaction will be equal 
to each other, whatever be the nature and motion of the 
system." That this statement of Newton's implicitly 
contains nearly the whole doctrine of energy was first 
pointed out by Thomson and Tait*. 

The words Action and Reaction as they occur in the 
enunciation of the Third Law of Motion are explained 
to mean Forces, that is to say, they are the opposite 
aspects of one and the same Stress. 

In the passage quoted above a new and different 
sense is given to these words by estimating Action and 
Reaction by the product of a force into the velocity of 

* Treatise on Natural Philosophy, vol. i, 1867, 268. 

"Newton, in a Scholium to his Third Law of Motion, has 
stated the relation between work and kinetic energy in a manner 
so perfect that it cannot be improved, but at the same time with 
so little apparent effort or desire to attract attention that no 
one seems to have been struck with the great importance of the 
passage till it was pointed out recently (1867) by Thomson and 
Tait." Clerk Maxwell's Theory of Heat, ch. iv on "Elementary 
Dynamical Principles," p. 91. 



76 ON ENERGY [CH. 

its point of application. According to this definition 
the Action of the external agent is the rate at which it 
does work. This is what is meant by the Power of a 
steam-engine or other prime mover. It is generally 
expressed by the estimated number of ideal horses 
which would be required to do the work at the same 
rate as the engine, and this is called the Horse-power 
of the engine. 

When we wish to express by a single word the rate 
at which work is done by an agent we shall call it the 
Power of the agent, defining the power as the work 
done in the unit of time. 

The use of the term Energy, in a precise and scientific 
sense, to express the quantity of work which a material 
system can do, was introduced by Dr Young*. 



97. ON THE DIFFERENT FORMS OF ENERGY 

The energy which a body has in virtue of its motion 
is called kinetic energy. 

A system may also have energy in virtue of its con- 
figuration, if the forces of the system are such that the 
system will do work against external resistance while it 
passes into another configuration. This energy is called 
Potential Energy. Thus when a stone has been lifted 
to a certain height above the earth's surface, the system 
of two bodies, the stone and the earth, has potential 
energy, and is able to do a certain amount of work 
during the descent of the stone. This potential energy 
is due to the fact that the stone and the earth attract 
each other, so that work has to be spent by the man 
who lifts the stone and draws it away from the earth, 
and after the stone is lifted the attraction between the 
earth and the stone is capable of doing work as the stone 
descends. This kind of energy, therefore, depends 
upon the work which the forces of the system would do 

* Lectures on Natural Philosophy [1807], Lecture VIII. 



v] DIFFERENT FORMS OF ENERGY 77 

if the parts of the system were to yield to the action 
of these forces. This is called the "Sum of the Ten- 
sions" by Helmholtz in his celebrated memoir on the 
" Conservation of Energy." * Thomson called it Statical 
Energy; it has also been called Energy of Position; 
but Rankine introduced the term Potential Energyf a 
very felicitous expression, since it not only signifies the 
energy which the system has not in actual possession, 
but only has the power to acquire, but it also indicates 
its connexion with what has been called (on other 
grounds) the Potential Function^. 

The different forms in which energy has been found 
to exist in material systems have been placed in one or 
other of these two classes Kinetic Energy, due to 
motion, and Potential Energy, due to configuration. 

Thus a hot body, by giving out heat to a colder body, 
may be made to do work by causing the cold body to 
expand in opposition to pressure. A material system, 
therefore, in which there is a non-uniform distribution 
of temperature has the capacity of doing work, or energy. 
This energy is now believed to be kinetic energy, due to 
a motion of agitation in the smallest parts of the hot body. 

Gunpowder has energy, for when fired it is capable 
of setting a cannon-ball in motion. The energy of gun- 
powder is Chemical Energy, arising from the power 
which the constituents of gunpowder possess of 
arranging themselves in a new manner when exploded, 
so as to occupy a much larger volume than the gun- 
powder does. In the present state of science chemists 
figure to themselves chemical action as a rearrangement 
of particles under the action of forces tending to produce 

* Berlin, 1847: translated in Taylor's Scientific Memoirs, Feb. 
1853. [Remarkable mainly for its wide ramifications into electric 
and chemical theory.] 

f The vis potentialis of Daniel Bernoulli, as contrasted with 
vis viva, e.g. for the case of a bent spring; cf. Euler, De Curvis 
Elasticis, in Appendix to Solutio Problematis I soperimetrici . . . 



{ The term Potential was employed independently by Gauss 
and by Green, and so probably originated with D. Bernoulli. 



7 8 ON ENERGY [CH.V 

this change of arrangement. From this point of view, 
therefore, chemical energy is potential energy. 

Air, compressed in the chamber of an air-gun, is 
capable of propelling a bullet. The energy of com- 
pressed air was at one time supposed to arise from the 
mutual repulsion of its particles. If this explanation 
were the true one its energy would be potential energy. 
In more recent times it has been thought that the 
particles of the air are in a state of motion, and that 
its pressure is caused by the impact of these particles 
on the sides of the vessel. According to this theory 
the energy of compressed air is kinetic energy. 

There are thus many different modes in which a 
material system may possess energy, and it may be 
doubtful in some cases whether the energy is of the 
kinetic or the potential form. The nature of energy, 
however, is the same in whatever form it may be found. 
The quantity of energy can always be expressed as 
equated to that of a body of a definite mass moving with 
a definite velocity. 



CHAPTER VI 
RECAPITULATION 

98. RETROSPECT OF ABSTRACT DYNAMICS 

WE have now gone through that part of the funda- 
mental science of the motion of matter which we have 
been able to treat in a manner sufficiently elementary 
to be consistent with the plan of this book. 

It remains for us to take a general view of the rela- 
tions between the parts of this science, and of the whole 
to other physical sciences, and this we can now do in 
a more satisfactory way than we could before we had 
entered into the subject. 

99. KINEMATICS 

We began with kinematics, or the science of pure 
motion. In this division of the subject the ideas brought 
before us are those of space and time. The only attri- 
bute of matter which comes before us is its continuity 
of existence in space and time the fact, namely, that 
every particle of matter, at any instant of time, is in 
one place and in one only, and that its change of place 
during any interval of time is accomplished by moving 
along a continuous path. 

Neither the force which affects the motion of the 
body, nor the mass of the body, on which the amount of 
force required to produce the motion depends, come 
under our notice in the pure science of motion. 

100. FORCE 

In the next division of the subject force is considered 
in the aspect of that which alters the motion of a mass. 

If we confine our attention to a single body, our in- 
vestigation enables us, from observation of its motion, to 



8o FORCE AND STRESS [CH. 

determine the direction and magnitude of the resultant 
force which acts on it, and this investigation is the 
exemplar and type of all researches undertaken for the 
purpose of the discovery and measurement of physical 
forces. 

But this may be regarded as a mere application of 
the definition of a force, and not as a new physical 
truth. 

It is when we come to define equal forces as those 
which produce equal rates of acceleration in the same 
mass, and equal masses as those which are equally 
accelerated by equal forces, that we find that these 
definitions of equality amount to the assertion of the 
physical truth, that the comparison of quantities of 
matter by the forces required to produce in them a given 
acceleration is a method which always leads to con- 
sistent results, whatever be the absolute values of the 
forces and the accelerations. 

101. STRESS 

The next step in the science of force is that in which 
we pass from the consideration of a force as acting on 
a body, to that of its being one aspect of that mutual 
action between two bodies, which is called by Newton 
Action and Reaction, and which is now more briefly 
expressed by the single word Stress. 

102. RELATIVITY OF DYNAMICAL KNOWLEDGE 

Our whole progress up to this point may be described 
as a gradual development of the doctrine of relativity of 
all physical phenomena. Position we must evidently 
acknowledge to be relative, for we cannot describe the 
position of a body in any terms which do not express 
relation. The ordinary language about motion and rest 
does not so completely exclude the notion of their being 
measured absolutely, but the reason of this is, that in 
our ordinary language we tacitly assume that the earth 
is at rest. 



vi] RELATIVITY OF FORCE 81 

As our ideas of space and motion become clearer, we 
come to see how the whole body of dynamical doctrine 
hangs together in one consistent system. 

Our primitive notion may have been that to know 
absolutely where we are, and in what direction we are 
going, are essential elements of our. knowledge as con- 
scious beings. 

But this notion, though undoubtedly held by many 
wise men in ancient times, has been gradually dispelled 
from the minds of students of physics. 

There are no landmarks in space; one portion of 
space is exactly like every other portion, so that we 
cannot tell where we are. We are, as it were, on an 
unruffled sea, without stars, compass, soundings, wind, 
or tide, and we cannot tell in what direction we are 
going. We have no log which we can cast out to take 
a dead reckoning by; we may compute our rate of 
motion with respect to the neighbouring bodies, but 
we do not know how these bodies may be moving in 
space. 

103. RELATIVITY OF FORCE 

We cannot even tell what force may be acting on us ; 
we can only tell the difference between the force acting 
on one thing and that acting on another*. 

We have an actual example of this in our every-day 
experience. The earth moves round the sun in a year 
at a distance of 91,520,000 miles or 1-473 x io 13 
centimetres-]-. It follows from this that a force is exerted 
on the earth in the direction of the sun, which produces 
an acceleration of the earth in the direction of the sun 
of about 0-019 m f eet an d seconds, or about y^ of the 
intensity of gravity at the earth's surface. 

A force equal to the sixteen-hundredth part of the 
weight of a body might be easily measured by known 
experimental methods, especially if the direction of this 

* See Appendix I; especially p. 143. 

t More modern values are 9-28 x io 7 miles, or 1-494 x io" cm. 



82 UNIVERSALITY OF GRAVITATION [CH. 

force were differently inclined to the vertical at different 
hours of the day. 

Now, if the attraction of the sun were exerted upon 
the solid part of the earth, as distinguished from the 
movable bodies on which we experiment, a body sus- 
pended by a string, and moving with the earth, would 
indicate the difference between the solar action on the 
body, and that on the earth as a whole. 

If, for example, the sun attracted the earth and not 
the suspended body, then at sunrise the point of sus- 
pension, which is rigidly connected with the earth, 
would be drawn towards the sun, while the suspended 
body would be acted on only by the earth's attraction, 
and the string would appear to be deflected away from 
the sun by a sixteen-hundredth part of the length of 
the string. At sunset the string would be deflected away 
from the setting sun by an equal amount; and as the 
sun sets at a different point of the compass from that 
at which he rises the deflexions of the string would be 
in different directions, and the difference in the position 
of the plumb-line at sunrise and sunset would be easily 
observed. 

But instead of this, the attraction of gravitation is 
exerted upon all kinds of matter equally at the same 
distance from the attracting body. At sunrise and 
sunset the centre of the earth and the suspended body 
are nearly at the same distance from the sun, and no 
deflexion of the plumb-line due to the sun's attraction 
can be observed at these times. The attraction of the 
sun, therefore, in so far as it is exerted equally upon all 
bodies on the earth, produces no effect on their relative 
motions. It is only the differences of the intensity and 
direction of the attraction acting on different parts of 
the earth which can produce any effect, and these 
differences are so small for bodies at moderate distances 
that it is only when the body acted on is very large, as 
in the case of the ocean, that their effect becomes per- 
ceptible in the form of tides. 



vi] DETERMINATION OF DIRECTION 83 

104. ROTATION 

In what we have hitherto said about the motion of 
bodies, we have tacitly assumed that in comparing one 
configuration of the system with another, we are able 
to draw a line in the final configuration parallel to a 
line in the original configuration. In other words, we 
assume that there are certain directions in space which 
may be regarded as constant, and to which other direc- 
tions may be referred during the motion of the system. 
In astronomy, a line drawn from the earth to a star 
may be considered as fixed in direction, because the 
relative motion of the earth and the star is in general 
so small compared with the distance between them that 
the change of direction, even in a century, is very small. 
But it is manifest that all such directions of reference 
must be indicated by the configuration of a material 
system existing in space, and that if this system were 
altogether removed, the original directions of reference 
could never be recovered. 

But though it is impossible to determine the absolute 
velocity of a body in space, it is possible to determine 
whether the direction of a line in a material system is 
constant or variable. 

For instance, it is possible by observations made on 
the earth alone, without reference to the heavenly bodies, 
to determine whether the earth is rotating or not. 

So far as regards the geometrical configuration of the 
earth and the heavenly bodies, it is evidently all the same* 
"Whether the sun, predominant in heaven, 
Rise on the earth, or earth rise on the sun; 
He from the east his flaming road begin, 
Or she from west her silent course advance 
With inoffensive pace that spinning sleeps 
On her soft axle, while she paces even, 
And bears thee soft with the smooth air along." 

* From the discussion on the celestial motions in Paradise Lost 
(Book vin, lines 160-6) : Milton's interview with Galileo when as 
a young man he visited Italy may be recalled. 

62 



84 RELATIVITY [CH. 

The distances between the bodies composing the 
universe, whether celestial or terrestrial, and the angles 
between the lines joining them, are all that can be 
ascertained without an appeal to dynamical principles, 
and these will not be affected if any motion of rotation 
of the whole system, similar to that of a rigid body 
about an axis, is combined with the actual motion; so 
that from a geometrical point of view the Copernican 
system, according to which the earth rotates, has no 
advantage, except that of simplicity, over that in which 
the earth is supposed to be at rest, and the apparent 
motions of the heavenly bodies to be their absolute 
motions. 

Even if we go a step further, and consider the dyna- 
mical theory of the earth rotating round its axis, we 
may account for its oblate figure, and for the equi- 
librium of the ocean and of all other bodies on its 
surface on either of two hypotheses that of the motion 
of the earth round its axis, or that of the earth not 
rotating, but caused to assume its oblate figure by a 
force acting outwards in all directions from its axis, the 
intensity of this force increasing as the distance from 
the axis increases. Such a force, if it acted on all 
kinds of matter alike, would account not only for the 
oblateness of the earth's figure, but for the conditions 
of equilibrium of all bodies at rest with respect to the 
earth. 

It is only when we go further still, and consider the 
phenomena of bodies which are in motion with respect 
to the earth*, that we are really constrained to admit 
that the earth rotates. 

105. NEWTON'S DETERMINATION OF THE ABSOLUTE 

VELOCITY OF ROTATION 

Newton was the first to point out that the absolute 
motion of rotation of the earth might be demonstrated 
by experiments on the rotation of a material system. 
* As in Art. 105. See also Appendix I, p. 142. 



vi] ABSOLUTE ROTATION 85 

For instance, if a bucket of water is suspended from a 
beam by a string, and the string twisted so as to keep 
the bucket spinning round a vertical axis, the water 
will soon spin round at the same rate as the bucket, so 
that the system of the water and the bucket turns round 
its axis like a solid body. 

The water in the spinning bucket rises up at the 
sides, and is depressed in the middle, showing that in 
order to make it move in a circle a pressure must be 
exerted towards the axis. This concavity of the surface 
depends on the absolute motion of rotation of the water 
and not on its relative rotation. 

For instance, it does not depend on the rotation 
relative to the bucket. For at the beginning of the 
experiment, when we set the bucket spinning, and 
before the water has taken up the motion, the water 
and the bucket are in relative motion, but the surface 
of the water is flat, because the water is not rotating, 
but only the bucket. 

When the water and the bucket rotate together, 
there is no motion of the one relative to the other, but 
the surface of the water is hollow, because it is rotating. 

When the bucket is stopped, as long as the water 
continues to rotate its surface remains hollow, showing 
that it is still rotating though the bucket is not. 

It is manifestly the same, as regards this experiment, 
whether the rotation be in the direction of the hands 
of a watch or the opposite direction, provided the rate 
of rotation is the same. 

Now let us suppose this experiment tried at the 
North Pole. Let the bucket be made, by a proper 
arrangement of clockwork, to rotate either in the direc- 
tion of the hands of a watch, or in the opposite direction, 
at a perfectly regular rate. 

If it is made to turn round by clockwork once in 
twenty-four hours (sidereal time) the way of the hands 
of a watch laid face upwards, it will be rotating as 
regards the earth, but not rotating as regards the stars. 



86 TEST OF EARTH'S ROTATION [CH. 

If the clockwork is stopped, it will rotate with 
respect to the stars, but not with respect to the earth. 

Finally, if it is made to turn round once in twenty- 
four hours (sidereal time) in the opposite direction, it 
will be rotating with respect to the earth at the same 
rate as at first, but instead of being free from rotation 
as respects the stars, it will be rotating at the rate of one 
turn in twelve hours. 

Hence if the earth is at rest, and the stars moving 
round it, the form of the surface will be the same in the 
first and last case; but if the earth is rotating, the 
water will be rotating in the last case but not in the 
first, and this will be made manifest by the water rising 
higher at the sides in the last case than in the first. 

The surface of the water will not be really concave 
in any of the cases supposed, for the effect of gravity 
acting towards the centre of the earth is to make the 
surface convex, as the surface of the sea is, and the 
rate of rotation in our experiment is not sufficiently 
rapid to make the surface concave. It will only make 
it slightly less convex than the surface of the sea in the 
last case, and slightly more convex in the first. 

But the difference in the form of the surface of 
the water would be so exceedingly small, that with our 
methods of measurement it would be hopeless to 
attempt to determine the rotation of the earth in this 
way. 

1 06. FOUCAULT'S PENDULUM 

The most satisfactory method of making an experi- 
ment for this purpose is that devised by M. Foucault*. 

A heavy ball is hung from a fixed point by a wire, so 
that it is capable of swinging like a pendulum in any 
vertical plane passing through the fixed point. 

* Nowadays the fixity of direction in space of the plane of 
rotation of a rapidly spinning wheel, freely pivoted, a method 
also originated by Foucault, would reveal it most readily. Cf. 
Art. 71. The gyrostatic compass interacts with the earth's rota- 
tion, on the same principle. 



vi] FOUCAULT'S PENDULUM 87 

In starting the pendulum care must be taken that 
the wire, when at the lowest point of the swing, passes 
exactly through the position it assumes when hanging 
vertically. If it passes on one side of this position, it 
will return on the other side, and this motion of the 
pendulum round the vertical instead of through the 
vertical must be carefully avoided, because we wish to 
get rid of all motions of rotation either in one direction 
or the other. 

Let us consider the angular momentum of the pen- 
dulum about the vertical line through the fixed point. 

At the instant at which the wire of the pendulum 
passes through the vertical line, the angular momentum 
about the vertical line is zero. 

The force of gravity always acts parallel to this 
vertical line, so that it cannot produce angular momen- 
tum round it. The tension of the wire always acts 
through the fixed point, so that it cannot produce 
angular momentum about the vertical line. 

Hence the pendulum can never acquire angular 
momentum about the vertical line through the point of 
suspension. 

Hence when the wire is out of the vertical, the 
vertical plane through the centre of the ball and the 
point of suspension cannot be rotating; for if it were, 
the pendulum would have an angular momentum about 
the vertical line*. 

Now let us suppose this experiment performed at 
the North Pole. The plane of vibration of the pendulum 
will remain absolutely constant in direction, so that if 
the earth rotates, the rotation of the earth will be made 
manifest. 

* But if from want of precaution the ball described an open 
elliptic curve, however elongated, this curve of vibration would 
rotate spontaneously, through an angle Ji2 in each revolution of 
the ball, and in the same direction, where is the (small) extent 
of the conical angle traced out by the wire. This may readily 
mask the effect of the earth's rotation. If the bob were free to 
revolve on the wire as axis, that body would turn through Q in 
each revolution. 



88 ENERGY IS FUNDAMENTAL [CH. 

We have only to draw a line on the earth parallel 
to the plane of vibration, and to compare the position 
of this line with that of the plane of vibration at a 
subsequent time. 

As a pendulum of this kind properly suspended will 
swing for several hours, it is easy to ascertain whether 
the position of the plane of vibration is constant as 
regards the earth, as it would be if the earth is at rest, 
or constant as regards the stars, if the stars do not move 
round the earth. 

We have supposed, for the sake of simplicity in the 
description, that the experiment is made at the North 
Pole. It is not necessary to go there in order to 
demonstrate the rotation of the earth. The only region 
where the experiment will not show it is at the equator. 

At every other place the pendulum will indicate the 
rate of rotation of the earth with respect to the vertical 
line at that place. If at any instant the plane of the 
pendulum passes through a star near the horizon either 
rising or setting, it will continue to pass through that 
star as long as it is near the horizon. That is to say, 
the horizontal part of the apparent motion of a star on 
the horizon is equal to the rate of rotation of the plane 
of vibration of the pendulum. 

It has been observed that the plane of vibration 
appears to rotate in the opposite direction in the 
southern hemisphere, and by a comparison of the rates 
at various places the actual time of rotation of the earth 
has been deduced without reference to astronomical 
observations. The mean value, as deduced from these 
experiments by Messrs Galbraith and Haughton in 
their Manual of Astronomy, is 23 hours 53 minutes 
37 seconds. The true time of rotation of the earth is 
23 hours 56 minutes 4 seconds mean solar time. 



vi] TEST OF A SUBSTANCE 89 

107. MATTER AND ENERGY* 

All that we know about matter relates to the series 
of phenomena in which energy is transferred from one 
portion of matter to another, till in some part of the 
series our bodies are affected, and we become conscious 
of a sensation. 

By the mental process which is founded on such 
sensations we come to learn the conditions of these 
sensations, and to trace them to objects which are not 
part of ourselves, but in every case the fact that we 
learn is the mutual action between bodies. This 
mutual action we have endeavoured to describe in this 
treatise. Under various aspects it is called Force, 
Action and Reaction, and Stress, and the evidence of 
it is the change of the motion of the bodies between 
which it acts. 

The process by which stress produces change of 
motion is called Work, and, as we have already shown, 
work may be considered as the transference of Energy 
from one body or system to another. 

Hence, as we have said, we are acquainted with 
matter only as that which may have energy communi- 
cated to it from other matter, and which may, in its 
turn, communicate energy to other matter. 

Energy, on the other hand, we know only as that 
which in all natural phenomena is continually passing 
from one portion of matter to another. 

1 08. TEST OF A MATERIAL SUBSTANCE 

Energy cannot exist except in connexion with matter. 
Hence since, in the space between the sun and the earth, 
the luminous and thermal radiations, which have left 
the sun and which have not reached the earth, possess 
energy, the amount of which per cubic mile can be 
measured, this energy must belong to matter existing 

* See Appendix II. 



90 ENERGY IS RELATIVE [CH. 

in the interplanetary spaces, and since it is only by the 
light which reaches us that we become aware of the 
existence of the most remote stars, we conclude that 
the matter which transmits light is disseminated through 
the whole of the visible universe. 

109. ENERGY NOT CAPABLE OF IDENTIFICATION 

We cannot identify a particular portion of energy, or 
trace it through its transformations. It has no individual 
existence, such as that which we attribute to particular 
portions of matter. 

The transactions of the material universe appear to 
be conducted, as it were, on a system of credit*. Each 
transaction consists of the transfer of so much credit 
or energy from one body to another. This act of 
transfer or payment is called work. The energy so 
transferred does not retain any character by which it 
can be identified when it passes from one form to 
another. 

no. ABSOLUTE VALUE OF THE ENERGY OF A BODY 
UNKNOWN 

The energy of a material system can only be esti- 
mated in a relative manner. 

In the first place, though the energy of the motion 
of the parts relative to the centre of mass of the system 
may be accurately defined, the whole energy consists 
of this together with the energy of a mass equal to that 
of the whole system moving with the velocity of the 
centre of mass. Now this latter velocity that of the 
centre of mass can be estimated only with reference to 
some body external to the system, and the value which 
we assign to this velocity will be different according to 
the body which we select as our origin. 

Hence the estimated kinetic energy of a material 

* Except perhaps that credit can be artificially increased, or 
inflated. 



vi] CONNECTING PRINCIPLE IN PHYSICS 91 

system contains a part, the value of which cannot be 
determined except by the arbitrary selection of an 
origin. The only origin which would not be arbitrary 
is the centre of mass of the material universe, but this 
is a point the position and motion of which are quite 
unknown to us. 

in. LATENT ENERGY 

But the energy of a material system is indeterminate 
for another reason. We cannot reduce the system to a 
state in which it has no energy, and any energy which 
is never removed from the system must remain un- 
perceived Jby us, for it is only as it enters or leaves the 
system that we can take any account of it. 

We must, therefore, regard the energy of a material 
system as a quantity of which we may ascertain the 
increase or diminution as the system passes from one 
definite condition to another. The absolute value of 
the energy in the standard condition is unknown to 
us, and it would be of no value to us if we did know it, 
as all phenomena depend on the variations of the energy, 
and not on its absolute value. 

112. A COMPLETE DISCUSSION OF ENERGY WOULD 

INCLUDE THE WHOLE OF PHYSICAL SCIENCE 

The discussion of the various forms of energy 
gravitational, electro-magnetic, molecular, thermal, etc. 
with the conditions of the transference of energy from 
one form to another, and the constant dissipation of the 
energy available for producing work, constitutes the 
whole of physical science, in so far as it has been de- 
veloped in the dynamical form under the various 
designations of Astronomy, Electricity, Magnetism, 
Optics, Theory of the Physical States of Bodies, 
Thermo-dynamics, and Chemistry. 



CHAPTER VII 



THE PENDULUM AND GRAVITY 

113. ON UNIFORM MOTION IN A CIRCLE 

LET M (fig. n) be a body moving in a circle with 
velocity V. 

Let OM = r be the radius of the circle. 

The direction of the velocity 
of M is that of the tangent to 
the circle. Draw OV parallel to 
this direction through the centre 
of the circle and equal to the 
distance described in unit of time 
with velocity V, then OV = V. 
If we take O as the origin of 
the diagram of velocity, V will 
represent the velocity of the 
body at M. 

As the body moves round the 
circle, the point V will also 

describe a circle, and the velocity of the point V will 
be to that of M as OV to OM. 

If, therefore, we draw OA in MO produced, and 
therefore parallel to the direction of motion of V, and 
make OA a third proportional to OM and OV, and 
if we assume O as the origin of the diagram of rate of 
acceleration, then the point A will represent the velocity 
of the point V, or, what is the same thing, the rate of 
acceleration of the point M. 

Hence, when a body moves with uniform velocity in 
a circle, its acceleration is directed towards the centre 
of the circle, and is a third proportional to the radius 
of the circle and the velocity of the body. 

The force acting on the body M is equal to the 




Fig. ii. 



CH.VII] CENTRIFUGAL FORCE 93 

product of this acceleration into the mass of the body, 
or if F be this force 



114. CENTRIFUGAL FORCE 

This force F is that which must act on the body M 
in order to keep it in the circle of radius r, in which 
it is moving with velocity V. 

The direction of this force is towards the centre of 
the circle. 

If this force is applied by means of a string fastened 
to the body, the string will be in a state of tension. 
To a person holding the other end of the string this 
tension will appear to be directed towards the body M, 
as if the body M had a tendency to move away from the 
centre of the circle which it is describing. 

Hence this latter force is often called Centrifugal 
Force. 

The force which really acts on the body, being directed 
towards the centre of the circle, is called Centripetal 
Force, and in some popular treatises the centripetal 
and centrifugal forces are described as opposing and 
balancing each other. But they are merely the different 
aspects of the same stress [acting in the string]. 

115. PERIODIC TIME 

The time of describing the circumference of the circle 
is called the Periodic Time. If TT represents the ratio 
of the circumference of a circle to its diameter, which 
is 3'i4i59. ., the circumference of a circle of radius r 
is 27rr ; and since this is described in the periodic time T 
with velocity V, we have 

27TT = VT. 

Hence F 



94 HARMONIC VIBRATIONS [CH. 

The rate of circular motion is often expressed by the 
number of revolutions in unit of time. Let this number 
[the frequency] be denoted by n, then 

nT= i 
and F= 



1 1 6. ON SIMPLE HARMONIC VIBRATIONS 

If while the body M (fig. n) moves in a circle with 
uniform velocity another point P moves in a fixed 
diameter of the circle, so as to be always at the foot 
of the perpendicular from M on that diameter, the 
body P is said to execute Simple Harmonic Vibrations. 

The radius, r, of the circle is called the Amplitude of 
the vibration. 

The periodic time of M is called the Periodic Time 
of Vibration. 

The angle which OM makes with the positive 
direction of the fixed diameter is called the Phase of 
the vibration. 

Iiy. ON THE FORCE ACTING ON THE 

VIBRATING BODY 

The only difference between the motions of M and 
P is that M has a vertical motion compounded with 
a horizontal motion which is the same as that of P. 
Hence the velocity and the acceleration of the two bodies 
differ only with respect to the vertical part of the 
velocity and acceleration of M. 

The acceleration of P is therefore the horizontal 
component of that of M, and since the acceleration 
of M is represented by OA, which is in the direction 
of MO produced, the acceleration of P will be repre- 
sented by OB, where B is the foot of the perpendicular 
from A on the horizontal diameter. Now by similar 
triangles OMP, OAB 

OM : OA = OP : OB. 



vn] ISOCHRONOUS 95 

But OM = r and OA = - 477* ^ 

Hence OB = - ^J OP - - 4 7r 2 2 OP. 

In simple harmonic vibration, therefore, the ac- 
celeration is always directed towards the centre of 
vibration, and is equal to the distance from that centre 
multiplied by 47r 2 w 2 , and if the mass of the vibrating 
body is P, the force acting on it at a distance x from 
O is 47r 2 w 2 P*. 

It appears, therefore, that a body which executes 
simple harmonic vibrations in a straight line is acted 
on by a force which varies as the distance from the 
centre of vibration, and the value of this force at a 
given distance depends only on that distance, on the 
mass of the body, and on the square of the number 
of vibrations in unit of time, and is independent of the 
amplitude of the vibrations. 



118. ISOCHRONOUS VIBRATIONS 

It follows from this that if a body moves in a straight 
line and is acted on by a force directed towards a fixed 
point on the line and varying as the distance from that 
point, it will execute simple harmonic vibrations, the 
periodic time of which will be the same whatever the 
amplitude of vibration. 

If for a particular kind of displacement of a body, 
as turning round an axis, the force tending to bring it 
back to a given position varies as the displacement, 
the body will execute simple harmonic vibrations 
about that position, the periodic time of which will be 
independent of their amplitude. 

Vibrations of this kind, which are executed in the 
same time whatever be their amplitude, are called 
Isochronous Vibrations. 



96 POTENTIAL ENERGY OF A VIBRATOR [CH. 

119. POTENTIAL ENERGY OF THE VIBRATING BODY 

The velocity of the body when it passes through the 
point of equilibrium is equal to that of the body moving 
in the circle, or 

V = 27rrn, 

where r is the amplitude of vibration and n is the 
number of double vibrations per second. 

Hence the kinetic energy of the vibrating body at 
the point of equilibrium is 



where M is the mass of the body. 

At the extreme elongation, where x = r, the velocity, 
and therefore the kinetic energy, of the body is zero. 
The diminution of kinetic energy must correspond to 
an equal increase of potential energy. Hence if we 
reckon the potential energy from the configuration in 
which the body is at its point of equilibrium, its 
potential energy when at a distance r from this point 
is 2 7T z Mn 2 r 2 . 

This is the potential energy of a body which vibrates 
isochronously, and executes n double vibrations per 
second when it is at rest at the distance, r, from the 
point of equilibrium. As the potential energy does not 
depend on the motion of the body, but only on its 
position, we may write it 

27T 2 M 2 * 2 , 

where x is the distance from the point of equilibrium. 

120. THE SIMPLE PENDULUM 

The simple pendulum consists of a small heavy body 
called the bob, suspended from a fixed point by a fine 
string of invariable length. The bob is supposed to 
be so small that its motion may be treated as that of a 
material particle, and the string is supposed to be so 



vn] TIME OF VIBRATION 97 

fine that we may neglect its mass and weight. The 
bob is set in motion so as to swing through a small 
angle in a vertical plane. Its path, therefore, is an arc 
of a circle, whose centre is the point of suspension, 
O, and whose radius is the length of the string, which 
we shall denote by /. 

Let O (fig. 12) be the point B 

of suspension and OA the 
position of the pendulum when 
hanging vertically. When the 
bob is at M it is higher than 
when it is at A by the height 

where AM is the 




M 



chord of the arc AM and 
AB = 2l. 

If M be the mass of the bob 
and g the intensity of gravity 
the weight of the bob will be Mg and the work done 
against gravity during the motion of the bob from A 
to M will be MgAP. This, therefore, is the potential 
energy of the pendulum when the bob is at M, reckon- 
ing the energy zero when the bob is at A. 

We may write this energy 



The potential energy of the bob when displaced 
through any arc varies as the square of the chord of 
that arc. 

If it had varied as the square of the arc itself in 
which the bob moves, the vibrations would have been 
strictly isochronous. As the potential energy varies 
more slowly than the square of the arc, the period of 
each vibration will be greater when the amplitude is 
greater. 

For very small vibrations, however, we may neglect 
the difference between the chord and the arc, and 



98 SIMPLE PENDULUM [CH. 

denoting the arc by x we may write the potential 



But we have already shown that in harmonic vibrations 
the potential energy is 27r 2 M 2 # 2 . 

Equating these two expressions and clearing fractions 

wefind 



where g is the intensity of gravity, TT is the ratio of the 
circumference of a circle to its diameter, n is the number 
of vibrations of the pendulum in unit of time, and / is 
the length of the pendulum. 



121. A RIGID PENDULUM 

If we could construct a pendulum with a bob so 
small and a string so fine that it might be regarded 
for practical purposes as a simple pendulum, it would 
be easy to determine g by this method. But all real 
pendulums have bobs of considerable size, and in 
order to preserve the length invariable the bob must be 
connected with the point of suspension by a stout rod, 
the mass of which cannot be neglected. It is always 
possible, however, to determine the length of a simple 
pendulum whose vibrations would be executed in the 
same manner as those of a pendulum of any shape. 

The complete discussion of this subject would lead 
us into calculations beyond the limits of this treatise. 
We may, however, arrive at the most important result 
without calculation as follows. 

The motion of a rigid body in one plane may be 
completely defined by stating the motion of its centre 
of mass, and the motion of the body round its centre 
of mass. 

The force required to produce a given change in the 
motion of the centre of mass depends only on the mass 
of the body (Art. 63). 



vn] SOLID PENDULUM 99 

The moment required to produce a given change of 
angular velocity about the centre of mass depends on the 
distribution of the mass, being greater the further the 
different parts of the body are from the centre of mass. 

If, therefore, we form a system of two particles 
rigidly connected, the sum of the masses being equal 
to the mass of a pendulum, their centre of mass coin- 
ciding with that of the pendulum, and their distances 
from the centre of mass being such that a couple 
of the same moment is required to produce a given 
rotatory motion about the centre of mass of the new 
system as about that of the pendulum, then the new 
system will for motions in a certain plane be dynamic- 
ally equivalent to the given pendulum, that is, if the 
two systems are moved in the same way the forces 
required to guide the motion will be equal. Since the 
two particles may have any ratio, provided the sum 
of their masses is equal to the mass of the pendulum, 
and since the line joining them may have any direction 
provided it passes through the centre of mass, we may 
arrange them so that one of the particles corresponds to 
any given point of the pendulum, ____ __ _ 

say, the point of suspension P 



_^ 

(fig. 13). The mass of this par- ^ ~ 

ticle and the position and mass 
of the other at Q will be determinate. The position of 
the second particle, Q, is called the Centre of Oscilla- 
tion. Now in the system of two particles, if one of 
them, P, is fixed and the other, Q, allowed to swing 
under the action of gravity, we have a simple pendulum. 
For one of the particles, P, acts as the point of suspen- 
sion, and the other, Q, is at an invariable distance from 
it, so that the connexion between them is the same as if 
they were united by a string of length / = PQ. 

Hence a pendulum of any form swings in exactly 
the same manner as a simple pendulum whose length 
is the distance from the centre of suspension to the 
centre of oscillation. 

72 



ioo CENTRE OF OSCILLATION [CH. 

122. INVERSION OF THE PENDULUM 

Now let us suppose the system of two particles 
inverted, Q being made the point of suspension and 
P being made to swing. We have now a simple pen- 
dulum of the same length as before. Its vibrations 
will therefore be executed in the same time. But it 
is dynamically equivalent to the pendulum suspended 
by its centre of oscillation. 

Hence if a pendulum be inverted and suspended by 
its centre of oscillation its vibrations will have the 
same period as before, and the distance between the 
centre of suspension and that of oscillation will be 
equal to that of a simple pendulum having the same 
time of vibration. 

It was in this way that Captain Kater determined 
the length of the simple pendulum which vibrates 
seconds. 

He constructed a pendulum which could be made to 
vibrate about two knife edges, on opposite sides of the 
centre of mass and at unequal distances from it. 

By certain adjustments, he made the time of vibra- 
tion the same whether the one knife edge or the other 
were the centre of suspension. The length of the 
corresponding simple pendulum was then found by 
measuring the distance between the knife edges. 

123. ILLUSTRATION OF KATER'S PENDULUM 

The principle of Kater's Pendulum may be illus- 
trated by a very simple and striking experiment. Take 
a flat board of any form (fig. 14), and drive a piece of 
wire through it near its edge, and allow it to hang in 
a vertical plane, holding the ends of the wire by the 
finger and thumb. Take a small bullet, fasten it to the 
end of a thread and allow the thread to pass over the 
wire, so that the bullet hangs close to the board. Move 
the hand by which you hold the wire horizontally in 
the plane of the board, and observe whether the board 




vn] MEASUREMENT OF GRAVITY 101 

moves forwards or backwards with respect to the bullet. 
If it moves forwards lengthen the string, if backwards 
shorten it till the bullet and the board move together. 
Now mark the point of the board opposite the centre 
of the bullet and fasten the string to the 
wire. You will find that if you hold the 
wire by the ends and move it in any 
manner, however sudden and irregular, 
in the plane of the board, the bullet will 
never quit the marked spot on the board. 

Hence this spot is called the centre of 
oscillation, because when the board is 
oscillating about the wire when fixed it 
oscillates as if it consisted of a single 
particle placed at the spot. Fi - J 4- 

It is also called the centre of percussion, because if 
the board is at rest and the wire is suddenly moved 
horizontally the board will at first begin to rotate about 
the spot as a centre. 

124. DETERMINATION OF THE INTENSITY OF GRAVITY 

The most direct method of determining g is, no 
doubt, to let a body fall and find what velocity it has 
gained in a second, but it is very difficult to make accu- 
rate observations of the motion of bodies when their 
velocities are so great as 981 centimetres per second, 
and besides, the experiment would have to be conducted 
in a vessel from which the air has been exhausted, as 
the resistance of the air to such rapid motion is very 
considerable, compared with the weight of the falling 
body. 

The experiment with the pendulum is much more 
satisfactory. By making the arc of vibration very small, 
the motion of the bob becomes so slow that the resist- 
ance of the air can have very little influence on the time 
of vibration. In the best experiments the pendulum 
is swung in an air-tight vessel from which the air is 
exhausted. 



102 MODE OF COUNTING VIBRATIONS [CH. 

Besides this, the motion repeats itself, and the pen- 
dulum swings to and fro hundreds, or even thousands, 
of times before the various resistances to which it is 
exposed reduce the amplitude of the vibrations till they 
can no longer be observed. 

Thus the actual observation consists not in watching 
the beginning and end of one vibration, but in deter- 
mining the duration of a series of many hundred 
vibrations, and thence deducing the time of a single 
vibration. 

The observer is relieved from the labour of counting 
the whole number of vibrations, and the measurement 
is made one of the most accurate in the whole range of 
practical science by the following method. 

125. METHOD OF OBSERVATION 

A pendulum clock is placed behind the experimental 
pendulum, so that when both pendulums are hanging 
vertically the bob, or some other part of the experi- 
mental pendulum, just hides a white spot on the clock 
pendulum, as seen by a telescope fixed at some distance 
in front of the clock. 

Observations of the transit of "clock stars" across 
the meridian are made from time to time, and from 
these the rate of the clock is deduced in terms of 
" mean solar time." 

The experimental pendulum is then set a swinging, 
and the two pendulums are observed through the 
telescope. Let us suppose that the time of a single 
vibration is not exactly that of the clock pendulum, but 
a little more. 

The observer at the telescope sees the clock pendulum 
always gaining on the experimental pendulum, till at 
last the experimental pendulum just hides the white 
spot on the clock pendulum as it crosses the vertical 
line. The time at which this takes place is observed 
and recorded as the First Positive Coincidence. 

The clock pendulum continues to gain on the other, 



vn] ERROR OF OBSERVATION 103 

and after a certain time the two pendulums cross the 
vertical line at the same instant in opposite directions. 
The time of this is recorded as the First Negative Coin- 
cidence. After an equal interval of time there will be a 
second positive coincidence, and so on. 

By this method the clock itself counts the number, N, 
of vibrations of its own pendulum between the coinci- 
dences. During this time the experimental pendulum 
has executed one vibration less than the clock. Hence 
the time of vibration of the experimental pendulum is 

N 
.,_ seconds of clock time. 

When there is no exact coincidence, but when the 
clock pendulum is ahead of the experimental pendulum 
at one passage of the vertical and behind at the next, 
a little practice on the part of the observer will enable 
him to estimate at what time between the passages the 
two pendulums must have been in the same phase. The 
epoch of coincidence can thus be estimated to a fraction 
of a second. 

126. ESTIMATION OF ERROR 

The experimental pendulum will go on swinging for 
some hours, so that the whole time to be measured may 
be ten thousand or more vibrations. 

But the error introduced into the calculated time of 
vibration 
the time 



, by a mistake even of a whole second in noting 
of vibration, may be made exceedingly small 



by prolonging the experiment. 
For if we ot 



observe the first and the nth coincidence, 
and find that they are separated by an interval of 
Af seconds of the clock, the experimental pendulum 
will have lost n vibrations, as compared with the clock, 
and will have made N-n vibrations in N seconds. 

N 
Hence the time of a single vibration is T -. __ 

seconds of clock time. 

Let us suppose, however, that by a mistake of a 



104 PERIOD OF PENDULUM [CH. vn 

second we note down the last coincidence as taking 
place N + i seconds after the first. The value of T as 
deduced from this result would be 



and the error introduced by the mistake of a second 
will be Ar , A T 

T - T= I 



+ i - n N-n 



If N is 10,000 and n is 100, a mistake of one second 
in noting the time of coincidence will alter the value of 
T only about one-millionth part of its value. 



CHAPTER VIII 

UNIVERSAL GRAVITATION 

127. NEWTON'S METHOD 

THE most instructive example of the method of dynami- 
cal reasoning is that by which Newton determined 
the law of the force with which the heavenly bodies act 
on each other. . 

The process of dynamical reasoning consists in 
deducing from the successive configurations of the 
heavenly bodies, as observed by astronomers, their 
velocities and their accelerations, and in this way 
determining the direction and the relative magnitude 
of the force which acts on them. 

Kepler had already prepared the way for Newton's 
investigation by deducing from a careful study of the 
observations of Tycho Brahe the three laws of planetary 
motion which bear his name. 

128. KEPLER'S LAWS 

Kepler's Laws are purely kinematical. They com- 
pletely describe the motions of the planets, but they say 
nothing about the forces by which these motions are 
determined. 

Their dynamical interpretation was discovered by 
Newton. 

The first and second laws relate to the motion of a 
single planet. 

Law I. The areas swept out by the vector drawn 
from the sun to a planet are proportional to the times 
of describing them. If h denotes twice the area swept 
out in unit of time, twice the area swept out in time t 
will be ht, and if P is the mass of the planet, Pht will 
be the mass-area, as defined in Article 68. Hence the* 
angular momentum of the planet about the sun, which 



io6 KEPLER'S FIRST LAW [CH. 

is the rate of change of the mass-area, will be Ph, a 
constant quantity. 

Hence, by Article 70, the force, if any, which acts 
on the planet must have no moment with respect to 
the sun, for if it had it would increase or diminish the 
angular momentum at a rate measured by the value of 
this moment. 

Hence, whatever be the force which acts on the planet, 
the direction of this force must always pass through 
the sun. 

129. ANGULAR VELOCITY 

Definition. The angular velocity of a vector is the 
rate at which the angle increases which it makes with 
a fixed vector in the plane of its motion. 

If o is the angular velocity of a vector, and r its length, 
the rate at which it sweeps out an area is ^cor 2 . Hence, 

and since h is constant, o>, the angular velocity of a 
planet's motion round the sun, varies inversely as the 
square of the distance from the sun. 

This is true whatever the law of force may be, pro- 
vided the force acting on the planet always passes 
through the sun. 

130. MOTION ABOUT THE CENTRE OF MASS 
Since the stress between the planet and the sun acts 
on both bodies, neither of them can remain at rest. 

The only point whose 

-p motion is not affected 

by the stress is the 
Fig. 15. centre of mass of the 

two bodies. 
If r is the distance SP (fig. 15), and if C is the centre 

of mass, SC = -~ and CP = -^ p. The angular 

S 2 r 2 PS 2 h 

momentum of P about C is Po> /c p . 2 = , PX2 . 

^o T r) ^o T fj 



vin] ORBITS AND HODOGRAPHS 107 



131. THE ORBIT 

We have already made use of diagrams of configura- 
tion and of velocity in studying the motion of a material 
system. These diagrams, however, represent only the 
state of the system at a given instant; and this state is 
indicated by the relative position of points corresponding 
to the bodies forming the system. 

It is often, however, convenient to represent in a 
single diagram the whole series of configurations or 
velocities which the system assumes. If we suppose 
the points of the diagram to move so as continu- 
ally to represent the state of the moving system, each 
point of the diagram will trace out a line, straight or 
curved. 

On the diagram of configuration, this line is called, 
in general, the Path of the body. In the case of the 
heavenly bodies it is often called the Orbit. 



132. THE HODOGRAPH 

On the diagram of velocity the line traced out by 
each moving point is called the Hodograph of the body 
to which it corresponds. 

The study of the Hodograph, as a method of investi- 
gating the motion of a body, was introduced by Sir 
W. R. Hamilton. The hodograph may be defined as 
the path traced out by the extremity of a vector which 
continually represents, in direction and magnitude, the 
velocity of a moving body. 

In applying the method of the hodograph to a planet, 
the orbit of which is in one plane, we shall find it con- 
venient to suppose the hodograph turned round its 
origin through a right angle, so that the vector of the 
hodograph is perpendicular instead of parallel to the 
velocity it represents. 



io8 



KEPLER'S SECOND LAW 



[CH. 



133. KEPLER'S SECOND LAW 

Law II. The orbit of a planet with respect to the 
sun is an ellipse, the sun being in one of the foci. 

Let APQB (fig. 1 6) be the elliptic orbit. Let S 
be the sun in one focus, and let H be the other focus. 

Produce SP to U, 



U 




so that S U is equal 
to the transverse 
axis AB, and join 
HU, then HU will 
be proportional 
and perpendicular 
to the velocity at P. 
For bisect HU 
in Z and join ZP; 
ZP will be a tan- 
gent to the ellipse 
at P; let SY be 
a perpendicular 
from S on this 
tangent. 

If v is the ve- 
locity at P, and h twice the area swept out in unit of 
time, h=vSY. 

Also if b is half the conjugate axis of the ellipse 



Now 
hence 



SY.HZ=b 2 . 






Hence HU is always proportional to the velocity, 
and it is perpendicular to its direction. Now SU is 
always equal to AB. Hence the circle whose centre is 
S and radius AB is the hodograph of the planet, H 
being the origin of the hodograph. 

The corresponding points of the orbit and the hodo- 



vin] LAW OF GRAVITATION 109 

graph are those which lie in the same straight line 
through S. 

Thus P corresponds to U and Q to V. 

The velocity communicated to the body during its 
passage from P to Q is represented by the geometrical 
difference between the vectors H U and HV, that is, by 
the line UV, and it is perpendicular to this arc of the 
circle, and is therefore, as we have already proved, 
directed towards S. 

If PQ is the arc described in [a very small] time, then 
UV represents the acceleration [of velocity in that time ;] 
and since UV is on a circle whose centre is S, UV will 
be a measure of the angular [movement in that time] of 
the planet about S. Hence the acceleration is propor- 
tional to the angular velocity, and this by Art. 129 is 
inversely as the square of the distance SP. Hence the 
acceleration of the planet is in the direction of the sun, 
and is inversely as the square of the distance from the 
sun. 

This, therefore, is the law according to which the 
attraction of the sun on a planet varies as the planet 
moves in its orbit and alters its distance from the sun. 



134. FORCE ON A PLANET 

As we have already shown, the orbit of the planet 
with respect to the centre of mass of the sun and 
planet has its dimensions in the ratio of S to S + P 
to those of the orbit of the planet with respect to the 
sun. 

If 2a and zb are the axes of the orbit of the planet 
with respect to the sun, the area is -nab, and if T is the 
time of going completely round the orbit, the value of 

h is 2?r ~~i . The velocity with respect to the sun is there- 
fore 7T ^ HU. 

lo 



no KEPLER'S THIRD LAW [CH. 

With respect to the centre of mass it is 



The total acceleration of the planet towards the centre 
of mass [in describing an arc PQ] is 

S TJV 
S+PTb U 

and the impulse on the planet whose mass is P is 
therefore sp 

STPTb UV ' 
Let t be the time of describing PQ, then twice the 



h a 2 b 

and UV = 2acot = 2a -% t = 477 ~- 2 t. 

Hence the force on the planet [being impulse divided 
by time] is sp ^ 



This then is the value of the stress or attraction 
between a planet and the sun in terms of their masses 
P and , their mean distance a, their actual distance r, 
and the periodic time T. 



135. INTERPRETATION OF KEPLER'S THIRD LAW 

To compare the attraction between the sun and 
different planets, Newton made use of Kepler's third 
law. 

Law III. The squares of the periodic times of differ- 
ent planets are proportional to the cubes of their mean 

a 3 c 

distances. In other words yr 2 is a constant, say 2 . 

SP i 
Hence F = c --. 



vin] UNIVERSAL GRAVITATION in 

In the case of the smaller planets their masses are 

o 

so small, compared with that of the sun, that 

p 

may be put equal to i , so that F = c -% or the 

attraction on a planet is proportional to its mass and 
inversely as the square of its distance. 



136. LAW OF GRAVITATION 

This is the most remarkable fact about the attraction 
of gravitation, that at the same distance it acts equally 
on equal masses of substances of all kinds. This is 
proved by pendulum experiments for the different 
kinds of matter at the surface of the earth. Newton 
extended the law to the matter of which the different 
planets are composed. 

It had been suggested, before Newton proved it, 
that the sun as a whole attracts a planet as a whole, 
and the law of the inverse square had also been pre- 
viously stated, but in the hands of Newton the doctrine 
of gravitation assumed its final form. 

Every portion of matter attracts every other portion 
of matter, and the stress between them is proportional to 
the product of their masses divided by the square of 
their distance. 

For if the attraction between a gramme of matter in 

the sun and a gramme of matter in a planet at distance 

Q 
r is -g where C is a constant, then if there are S grammes 

in the sun and P in the planet the whole attraction 
between the sun and one gramme in the planet will be 

CS 

a-, and the whole attraction between the sun and the 

r SP 

planet will be C ^-. 

Comparing this statement of Newton's "Law of 



ii2 MUTUAL GRAVITATION [CH. 

Universal Gravitation" with the value of F formerly 
obtained we find 

^SP SP a 3 



or ^ 2 a 3 =C(S+P)T 2 . 



137. AMENDED FORM OF KEPLER'S THIRD LAW 

Hence Kepler's Third Law must be amended thus: 

The cubes of the mean distances are as the squares 
of the times multiplied into the sum of the masses of 
the sun and the planet. 

In the case of the larger planets, Jupiter, Saturn, etc., 
the value of S + P is considerably greater than in 
the case of the earth and the smaller planets. Hence 
the periodic times of the larger planets should be some- 
what less than they would be according to Kepler's law, 
and this is found to be the case. 

In the following table the mean distances (a) of the 
planets are given in terms of the mean distance of the 
earth, and the periodic times (T) in terms of the sidereal 
year: 

Planet a T a 3 7 12 a 3 - T* 

Mercury 0-387098 0-24084 0-0580046 0-0580049 -0-0000003 

Venus 0-72333 0-61518 0-378451 0-378453 -0-000002 

Earth i -oooo r-ooooo i-ooooo i-ooooo 

Mars 1-52369 1-88082 3-53746 3-5374? -o-ooooi 

Jupiter 5-20278 11-8618 140-832 140-701 +0-131 

Saturn 9*53879 29-4560 867-914 867-658 +0-456 

Uranus 19-1824 84-0123 7058-44 7058-07 +0-37 

Neptune 30-037 164-616 27100-0 27098-4 +1-6 

It appears from the table that Kepler's third law is 
very nearly accurate, for a 3 is very nearly equal to T 2 , 
but that for those planets whose mass is less than that 
of the earth namely, Mercury, Venus, and Mars a s 
is less than T 2 , whereas for Jupiter, Saturn, Uranus, 
and Neptune, whose mass is greater than that of the 
earth, a 3 is greater than T 2 . 



vm] ITS POTENTIAL ENERGY 113 

138. POTENTIAL ENERGY DUE TO GRAVITATION 

The potential energy of the gravitation between the 
bodies S and P may be calculated when we know the 
attraction between them in terms of their distance. 
The process of calculation by which we sum up the 
effects of a continually varying quantity belongs to the 
Integral Calculus, and though in this case the calcula- 
tion may be explained by elementary methods, we shall 
rather deduce the potential energy directly from Kepler's 
first and second laws. 

These laws completely define the motion of the sun 
and planet, and therefore we may find the kinetic energy 
of the system corresponding to any part of the elliptic 
orbit. Now, since the sun and planet form a conserva- 
tive system, the sum of the kinetic and potential energies 
is constant, and therefore when we know the kinetic 
energy we may deduce that part of the potential 
energy which depends on the distance between the 
bodies. 

139. KINETIC ENERGY OF THE SYSTEM 

To determine the kinetic energy we observe that the 
velocity of the planet with respect to the sun is by 
Article 133 , 



The velocities of the planet and the sun with respect 
to the centre of mass of the system are respectively 

s and P 

The kinetic energies of the planet and the sun are 
therefore pz 

? - and VSp+pp* 

and the whole kinetic energy is 

i SP i SP h 2 



ii4 KINETIC ENERGY IN ORBITS [CH. 

To determine [more directly] v 2 in terms of SP or r, 
we observe that by the law of areas 



also by a property of the ellipse 

HZ.SY=b 2 ...... (2), 

and by the similar triangles HZP and SYP 

SY_HP_ r 

HZ ~~ SP ~ 2 a - r 
multiplying (2) and (3) we find 



Hence by (i) 



and the kinetic energy of the system is 
SP (i i 



S+P 

and this by the equation at the end of Article 136 
becomes 



where C is the constant of gravitation. 

This is the value of the kinetic energy of the two 
bodies S and P when moving [relatively] in an ellipse 
of which the transverse axis is 2a. 

140. POTENTIAL ENERGY OF THE SYSTEM 
The sum of the kinetic and potential energies is 
constant, but its absolute value is by Article no un- 
known, and not necessary to be known. 

Hence if we [conclude, in accordance with the con- 
stancy of the total energy,] that the potential energy is 
of the form T 

K - C . SP - 



vni] MOTION OF THE MOON 115 

the second term, which is the only one depending on 
the distance, r, is also the only one which we have 
anything to do with. The. other term K represents the 
work done by gravitation while the two bodies originally 
at an infinite distance from each other are allowed 
to approach as near as their dimensions will allow them. 

141. THE MOON is A HEAVY BODY 

Having thus determined the law of the force between 
each planet and the sun, Newton proceeded to show 
that the observed weight of bodies at the earth's surface 
and the force which retains the moon in her orbit round 
the earth are related to each other according to the 
same law of the inverse square of the distance. 

This force of gravity acts in every region accessible 
to us, at the top of the highest mountains and at the 
highest point reached by balloons. Its intensity, as 
measured by pendulum experiments, decreases as we 
ascend ; and although the height to which we can ascend 
is so small compared with the earth's radius that we 
cannot from observations of this kind infer that gravity 
varies inversely as the square of the distance from the 
centre of the earth, the observed decrease of the inten- 
sity of gravity is consistent with this law, the form of 
which had been suggested to Newton by the motion of 
the planets. 

Assuming, then, that the intensity of gravity varies 
inversely as the square of the distance from the centre 
of the earth, and knowing its value at the surface of the 
earth, Newton calculated its value at the mean distance 
of the moon. 

His first calculations were vitiated by his adopting 
an erroneous estimate of the dimensions of the earth. 
When, however, he had obtained a more correct value 
of this quantity * he found that the intensity of gravity 

* And had demonstrated with great mathematical power the 
proposition assumed above, that the gravitation to a globe like 
the earth is exactly the same at all external points as if its mass 
were condensed to a point at its centre. 

82 



n6 LOCAL GRAVITATION [CH. 

calculated for a distance equal to that of the moon was 
equal to the force required to keep the moon in her 
orbit. 

He thus identified the force which acts between the 
earth and the moon with that which causes bodies near 
the earth's surface to fall towards the earth. 

142. CAVENDISH'S EXPERIMENT 

Having thus shown that the force with which the 
heavenly bodies attract each other is of the same kind 
as that with which bodies that we can handle are 
attracted to the earth, it remained to be shown that 
bodies such as we can handle attract one another. 

The difficulty of doing this arises from the fact that 
the mass of bodies which we can handle is so small 
compared with that of the earth, that even when we 
bring the two bodies as near as we can the attraction 
between them is an exceedingly small fraction of the 
weight of either. 

We cannot get rid of the attraction of the earth, but 
we must arrange the experiment in such a way that it 
interferes as little as is possible with the effects of the 
attraction of the other body. 

The apparatus devised by the Rev. John Michell * for 
this purpose was that which has since received the name 
of the Torsion Balance. Michell died before he was able 
to make the experiment, but his apparatus afterwards 
came into the hands of Henry Cavendishf, who im- 
proved it in many respects, and measured the attraction 
between [fixed] leaden balls and small balls suspended 
from the arms of the balance. A similar instrument 
was afterwards independently invented by Coulomb 
for measuring small electric and magnetic forces, and it 
continues to be the best instrument known to science 
for the measurement of small forces of all kinds. 

* Of Queens' College, Cambridge, Woodwardian Professor of 
Geology, 1762-4. See Memoir by Sir A. Geikie, Cambridge, 1918. 

f Of Peterhouse, Cambridge. See his Scientific Writings, 
2 vols., Cambridge, 1920. 



vm] PRINCIPLE OF TORSION ROD 117 

143. THE TORSION BALANCE* 

The Torsion Balance consists of a horizontal rod 
suspended by a wire from a fixed support. When 
the rod is turned round by an external force in a 
horizontal plane it twists the wire, and the wire being 
elastic tends to resist this strain and to untwist itself. 
This force of torsion is proportional to the angle 
through which the wire is twisted, so that if we cause 
a force to act in a horizontal direction at right angles 
to the rod at its extremity, we may, by observing the 
angle through which the force is able to turn the rod, 
determine the magnitude of the force. 

The force is proportional to the angle of torsion and 
to the fourth power of the diameter of the wire and in- 
versely to the length of the rod and the length of the wire. 

Hence, by using a long fine wire and a long rod, 
we may measure very small forces. 

In the experiment of Cavendish two spheres of equal 
mass, m, are suspended from the ex- 
tremities of the rod of the torsion 
balance. We shall for the present 
neglect the mass of the rod in com- 
parison with that of the spheres. Two 
larger spheres of equal mass, Af, are 
so arranged that they can be placed 
either at M and M or at M' and M'. 
In the former position they tend by 
their attraction on the smaller spheres, 
m and m, to turn the rod of the balance 
in the direction towards them. In the 
latter position they thus tend to turn it 
in the opposite direction. The torsion 
balance and its suspended spheres are 
enclosed in a case, to prevent their Fig I? 

being disturbed by currents of air. 
The position of the rod of the balance is ascertained 
* See infra, p. 143. 



n8 EXPERIMENTAL DETERMINATION [CH. 

by observing a graduated scale as seen by reflexion in 
a vertical mirror fastened to the middle of the rod. 
The balance is placed in a room by itself, and the 
observer does not enter the room, but observes the 
image of the graduated scale with a telescope. 

144. METHOD OF THE EXPERIMENT 

The time, T, of a double vibration of the torsion 
balance is first ascertained, and also the position of 
equilibrium of the centres of the suspended spheres. 

The large spheres are then brought up to the posi- 
tions M , M, so that the centre of each is at a distance 
from the position of equilibrium of the centre of the 
suspended sphere. 

No attempt is made to wait till the vibrations of the 
beam have subsided, but the scale-divisions corre- 
sponding to the extremities of a single vibration are 
observed, and are found to be distant x and y respec- 



x y 
Fig. 18. 



tively from the position of equilibrium. At these points 
the rod is, for an instant, at rest, so that its energy is 
entirely potential, and since the total energy is constant, 
the potential energy corresponding to the position x 
must be equal to that corresponding to the position y. 
Now if T be the time of a double vibration about the 
point of equilibrium o, the potential energy due to 
torsion when the scale reading is x is by Article 119 



and that due to the gravitation between m and M is by 
Article 140 



vni] THE CONSTANT OF GRAVITATION 119 

The potential energy of the whole system in the position 
x is therefore 



In the position y it is 



and since the potential energy in these two positions 
is equal, 



a~y a-x 
Hence ~, 27r 



By this equation C, the constant of gravitation, is 
determined in terms of the observed quantities, M the 
mass of the large spheres in grammes, T the time of a 
double vibration in seconds, and the distances x y y and 
a in centimetres. 

According to Baily's experiments, C = 6-5 x io~ 8 . 
If we assume the unit of mass, so that at a distance 
unity it would produce an acceleration unity, the centi- 
metre and the second being units, the unit of mass 
would be about 1-537 x io 7 grammes, or 15*37 tonnes. 
This unit of mass reduces C, the constant of gravitation, 
to unity. It is therefore used in the calculations of 
physical astronomy. 

145. UNIVERSAL GRAVITATION 

We have thus traced the attraction of gravitation 
through a great variety of natural phenomena, and have 
found that the law established for the variation of the 
force at different distances between a planet and the 
sun also holds when we compare the attraction between 
different planets and the sun, and also when we compare 
the attraction between the moon and the earth with that 
between the earth and heavy bodies at its surface. We 



120 UNIVERSAL GRAVITATION [CH. 

have also found that the gravitation of equal masses at 
equal distances is the same whatever be the nature of 
the material of which the masses consist. This we 
ascertain by experiments on pendulums of different 
substances, and also by a comparison of the attraction 
of the sun on different planets, which are probably 
not alike in composition. The experiments of Baily* 
on spheres of different substances placed in the 
torsion balance confirm this law. 

Since, therefore, we find in so great a number of 
cases occurring in regions remote from each other that 
the force of gravitation depends on the mass of bodies 
only, and not on their chemical nature or physical state, 
we are led to conclude that this is true for all substances. 

For instance, no man of science doubts that two 
portions of atmospheric air attract one another, although 
we have very little hope that experimental methods 
will ever be invented so delicate as to measure or even 
to make manifest this attraction. But we know that 
there is attraction between any portion of air and the 
earth, and we find by Cavendish's experiment that 
gravitating bodies, if of sufficient mass, gravitate 
sensibly towards each other, and we conclude that 
two portions of air gravitate towards each other. But 
it is still extremely doubtful whether the medium of 
light and electricity is a gravitating substance, though 
it is certainly material and has massf. 

146. CAUSE OF GRAVITATION 

Newton, in his Principia, deduces from the observed 
motions of the heavenly bodies the fact that they attract 
one another according to a definite law. 

* And more recently with extreme refinement by v. Jolly, Boys, 
Eotvos, and many others. Apparent weight is gravitation di- 
minished by centrifugal reaction to the earth's rotation: if these 
did not vary in the same way for all kinds of matter, delicate 
weighings would detect the discrepancy: the experiments of 
Eotvos show that it could not exceed five parts in io 8 . See infra, 

P- 143- 

j- See infra, p. 144. 



vin] NEWTON'S EXPOSITION 121 

This he gives as a result of strict dynamical reasoning, 
and by it he shows how not only the more conspicuous 
phenomena, but all the apparent irregularities of the 
motions of these bodies are the calculable results of 
this single principle. In his Principia he confines 
himself to the demonstration and development of this 
great step in the science of the mutual action of bodies. 
He says nothing about the means by which bodies are 
made to gravitate towards each other. We know that 
his mind did not rest at this point that he felt that 
gravitation itself must be capable of being explained, 
and that he even suggested an explanation depending 
on the action of an etherial medium pervading space. 
But with that wise moderation which is characteristic 
of all his investigations, he distinguished such specula- 
tions from what he had established by observation and 
demonstration, and excluded from his Principia all 
mention of the cause of gravitation, reserving his 
thoughts on this subject for the "Queries" printed at 
the end of his Opticks. 

The attempts which have been made since the time 
of Newton to solve this difficult question are few in 
number, and have not led to any well-established 
result*. 

147. APPLICATION OF NEWTON'S METHOD OF 
INVESTIGATION 

The method of investigating the forces which act 
between bodies which was thus pointed out and exem- 
plified by Newton in the case of the heavenly bodies, 
was followed out successfully in the case of electrified 
and magnetized bodies by Cavendish, Coulomb, and 
Poisson. 

The investigation of the mode in which the minute 
particles of bodies act on each other is rendered more 
difficult from the fact that both the bodies we consider 

* See Appendix I, infra, p. 140. 



122 MOLECULAR FORCES [CH.VIII 

and their distances are so small that we cannot perceive 
or measure them, and we are therefore unable to 
observe their motions as we do those of planets, or of 
electrified and magnetized bodies. 

148. METHODS OF MOLECULAR INVESTIGATIONS 

Hence the investigations of molecular science have 
proceeded for the most part by the method of hypo- 
thesis, and comparison of the results of the hypothesis 
with the observed facts. 

The success of this method depends on the generality 
of the hypothesis we begin with. If our hypothesis is 
the extremely general one that the phenomena to be 
investigated depend on the configuration and motion of 
a material system, then if we are able to deduce any 
available results from such an hypothesis, we may 
safely apply them to the phenomena before us*. 

If, on the other hand, we frame the hypothesis that 
the configuration, motion, or action of the material 
system is of a certain definite kind, and if the results 
of this hypothesis agree with the phenomena, then, 
unless we can prove that no other hypothesis would 
account for the phenomena, we must still admit the 
possibility of our hypothesis being a wrong one. 

149. IMPORTANCE OF GENERAL AND ELEMENTARY 

PROPERTIES 

It is therefore of the greatest importance in all 
physical inquiries that we should be thoroughly 
acquainted with the most general properties of material 
systems, and it is for this reason that in this book I 
have rather dwelt on these general properties than 
entered on the more varied and interesting field of the 
special properties of particular forms of matter. 

* This is the subject of the next chapter. 



[CHAPTER IX] 

ON THE EQUATIONS OF MOTION OF A 
CONNECTED SYSTEM* 

1. IN the fourth section of the second part of his 
Mecanique Analytique, Lagrange has given a method 
of reducing the ordinary dynamical equations of the 
motion of the parts' of a connected system to a number 
equal to that of the degrees of freedom of the system. 

The equations of motion of a connected system have 
been given in a different form by Hamilton, and have 
led to a great extension of the higher part of pure 
dynamics 1 . 

As we shall find it necessary, in our endeavours to 
bring electrical phenomena within the province of 
dynamics, to have our dynamical ideas in a state fit for 
direct application to physical questions, we shall devote 
this chapter to an exposition of these dynamical ideas 
from a physical point of view. 

2. The aim of Lagrange was to bring dynamics under 
the power of the calculus. He began by expressing the 
elementary dynamical relations in terms of the corre- 
sponding relations of pure algebraical quantities, and 
from the equations thus obtained he deduced his final 
equations by a purely algebraical process. Certain 
quantities (expressing the reactions between the parts 
of the system called into play by its physical connexions) 
appear in the equations of motion of the component 
parts of the system, and Lagrange's investigation, as 
seen from a mathematical point of view, is a method of 
eliminating these quantities from the final equations. 

In following the steps of this elimination the mind is 
exercised in calculation, and should therefore be kept 

* This chapter, now added, is a reprint of Part IV, Chapter v. 
of Maxwell's Treatise on Electricity and Magnetism (1873). 

1 See Professor Cay ley's "Report on Theoretical Dynamics," 
British Association, 1857; and Thomson and Tait's Natural Philo- 
sophy [1867]. 



124 MOTION OF A CONNECTED SYSTEM [CH. 

free from the intrusion of dynamical ideas. Our aim, 
on the other hand, is to cultivate our dynamical ideas. 
We therefore avail ourselves of the labours of the 
mathematicians, and retranslate their results from the 
language of the calculus into the language of dynamics, 
so that our words may call up the mental image, not 
of some algebraical process, but of some property of 
moving bodies. 

The language of dynamics has been considerably 
extended by those who have expounded in popular 
terms the doctrine of the Conservation of Energy, and 
it will be seen that much of the following statement is 
suggested by the investigation in Thomson and Tait's 
Natural Philosophy, especially the method of beginning 
with the theory of impulsive forces. 

I have applied this method so as to avoid the explicit 
consideration of the motion of any part of the system 
except the coordinates or variables, on which the 
motion of the whole depends. It is doubtless important 
that the student should be able to trace the connexion 
of the motion of each part of the system with that of 
the variables, but it is by no means necessary to do this 
in the process of obtaining the final equations, which are 
independent of the particular form of these connexions. 

The Variables 

3. The number of degrees of freedom of a system is 
the number of data which must be given in order 
completely to determine its position. Different forms 
may be given to these data, but their number depends 
on the nature of the system itself, and cannot be altered. 

To fix our ideas we may conceive the system con- 
nected by means of suitable mechanism with a number 
of moveable pieces, each capable of motion along a 
straight line, and of no other kind of motion. The 
imaginary mechanism which connects each of these 
pieces with the system must be conceived to be free 
from friction, destitute of inertia, and incapable of 



ix] THE VARIABLES 125 

being strained by the action of the applied forces. The 
use of this mechanism is merely to assist the imagination 
in ascribing position, velocity, and momentum to what 
appear, in Lagrange's investigation, as pure algebraical 
quantities. 

Let q denote the position of one of the moveable 
pieces as defined by its distance from a fixed point in 
its line of motion. We shall distinguish the values of 
q corresponding to the different pieces by the suffixes 
i, 2 , etc. When we are dealing with a set of quantities 
belonging to one piece only we may omit the suffix. 

When the values of all the variables (q) are given, the 
position of each of the moveable pieces is known, and, 
in virtue of the imaginary mechanism, the configuration 
of the entire system is determined. 

The Velocities 

4. During the motion of the system the configuration 
changes in some definite manner, and since the con- 
figuration at each instant is fully defined by the values 
of the variables (<?), the velocity of every part of the 
system, as well as its configuration, will be completely 
defined if we know the values of the variables (q), 



together with their velocities ^, or, according to 
Newton's notation, q). 

The Forces 

5. By a proper regulation of the motion of the vari- 
ables, any motion of the system, consistent with the 
nature of the connexions, may be produced. In order 
to produce this motion by moving the variable pieces, 
forces must be applied to these pieces. 

We shall denote the force which must be applied to 
any variable q r by F r . The system of forces (F) is 
mechanically equivalent (in virtue of the connexions 
of the system) to the system of forces, whatever it may 
be, which really produces the motion. 



126 MOTION OF A CONNECTED SYSTEM [CH. 

The Momenta 

6. When a body moves in such a way that its con- 
figuration, with respect to the force which acts on it, 
remains always the same (as, for instance, in the case 
of a force acting on a single particle in the line of its 
motion), the moving force is measured by the rate of 
increase of the momentum. If F is the moving force, 
and p the momentum, 



whence p = | Fdt. 

The time-integral of a force is called the Impulse of 
the force ; so that we may assert that the momentum is 
the impulse of the force which would bring the body 
from a state of rest into the given state of motion. 

In the case of a connected system in motion, the 
configuration is continually changing at a rate depending 
on the velocities (q), so that we can no longer assume 
that the momentum is the time-integral of the force 
which acts on it. 

But the increment Sq of any variable cannot be 
greater than q'&t, where 8* is the time during which the 
increment takes place, and q' is the greatest value of the 
velocity during that time. In the case of a system 
moving from rest under the action of forces always in 
the same direction, this is evidently the final velocity. 

If the final velocity and configuration of the system 
are given, we may conceive the velocity to be communi- 
cated to the system in a very small time Sf, the original 
configuration differing from the final configuration by 
quantities Sq^, 8q 2 , etc., which are less than qfit, 
q 2 St, etc. respectively. 

The smaller we suppose the increment of time 8t, 
the greater must be the impressed forces, but the time- 
integral, or impulse, of each force will remain finite. 
The limiting value of the impulse, when the time is 



ix] IMPULSE AND MOMENTUM 127 

diminished and ultimately vanishes, is defined as the 
instantaneous impulse; and the momentum p, corre- 
sponding to any variable q, is defined as the impulse 
corresponding to that variable, when the system is 
brought instantaneously from a state of rest into the 
given state of motion. 

This conception, that the momenta are capable of 
being produced by instantaneous impulses on the system 
at rest, is introduced only as a method of defining the 
magnitude of the momenta; for the momenta of the 
system depend only on the instantaneous state of motion 
of the system, and not on the process by which that 
state was produced. 

In a connected system the momentum corresponding 
to any variable is in general a linear function of the 
velocities of all the variables, instead of being, as in 
the dynamics of a particle, simply proportional to the 
velocity. 

The impulses required to change the velocities of the 
system suddenly from q lt q 2 , etc. to (?/, q 2 ', etc. are 
evidently equal to // p lt p z ' p 2 , etc. the changes 
of momentum of the several variables. 

Work done by a Small Impulse 

7. The work done by the force F 1 during the impulse 
is the space-integral of the force, or 

W= ! F,da, 



If <?/ is the greatest and <?/' the least value of the 
velocity q-^ during the action of the force, W must be 
less than r 

<?/ Fdt or <?i_' (PI Pi), 

and greater than 

Fdt or ji'fa'.-pj. 



128 MOTION OF A CONNECTED SYSTEM [CH. 

If we now suppose the impulse Fdt to be diminished 

without limit, the values of fa' and fa" will approach 
and ultimately coincide with that of fa, and we may 
write Pi p = &pi ; so that the work done is ultimately 



or, the work done by a very small impulse is ultimately the 
product of the impulse and the velocity. 

Increment of the Kinetic Energy 

8. When work is done in setting a conservative 
system in motion, energy is communicated to it, and 
the system becomes capable of doing an equal amount 
of work against resistances before it is reduced to rest. 

The energy which a system possesses in virtue of its 
motion is called its Kinetic Energy, and is communicated 
to it in the form of the work done by the forces which 
set it in motion. 

If T be the kinetic energy of the system, and if it 
becomes T+ S7 1 , on account of the action of an in- 
finitesimal impulse whose components are 8p lt 8/> 2 , etc. 
the increment 8J 1 must be the sum of the quantities 
of work done by the components of the impulse, or in 
symbols, _ 



The instantaneous state of the system is completely 
defined if the variables and the momenta are given. 
Hence the kinetic energy, which depends on the 
instantaneous state of the system, can be expressed in 
terms of the variables (q), and the momenta (/>). This 
is the mode of expressing T introduced by Hamilton. 
When T is expressed in this way we shall distinguish 
it by the suffix v , thus, T v . 

The complete variation of T v is 



ix] IN TERMS OF KINETIC ENERGY 129 

The last term may be written 



which diminishes with St, and ultimately vanishes [com- 
pared with the first term] when the impulse becomes 
instantaneous. 

Hence, equating the coefficients of 8p in equations (i) 
and (2), we obtain ^ T 



or, the velocity corresponding to the variable q is the 
differential coefficient of T v with respect to the corre- 
sponding momentum p. 

We have arrived at this result by the consideration of 
impulsive forces. By this method we have avoided the 
consideration of the change of configuration during the 
action of the forces. But the instantaneous state of the 
system is in all respects the same, whether the system was 
brought from a state of rest to the given state of motion by 
the transient application of impulsive forces, or whether 
it arrived at that state in any manner, however gradual. 

In other words, the variables, and the corresponding 
velocities and momenta, depend on the actual state of 
motion of the system at the given instant, and not on 
its previous history. 

Hence, the equation (3) is equally valid, whether the 
state of motion of the system is supposed due to impul- 
sive forces, or to forces acting in any manner whatever. 

We may now therefore dismiss the consideration of 
impulsive forces, together with the limitations imposed 
on their time of action, and on the changes of configura- 
tion during their action. 

Hamilton's Equations of Motion 
9. We have already shown that 



130 MOTION OF A CONNECTED SYSTEM [CH. 

Let the system move in any arbitrary way, subject to 
the conditions imposed by its connexions; then the 
variations of p and q are 

8?- $8* ...... (5). 



=!* 

and the complete variation of T v is 

***) 



But the increment of the kinetic energy arises from 
the work done by the impressed forces, or 

$T v =X(FSq) ...... (8). 

In these two expressions the variations 8q are all 
independent of each other, so that we are entitled to 
equate the coefficients of each of them in the two 
expressions (7) and (8). We thus obtain 



where the momentum p r and the force F r belong to the 
variable q r *. 

There are as many equations of this form as there are 
variables. These equations were given by Hamilton. 
They show that the force corresponding to any variable 
is the sum of two parts. The first part is the rate 
of increase of the momentum of that variable with 
respect to the time. The second part is the rate of in- 
crease of the kinetic energy per unit of increment of 
the variable, the other variables and all the momenta 
being constant. 

* But see infra, p. 158. 



ix] IN TERMS OF MOMENTA 131 

The Kinetic Energy expressed in Terms of the 
Momenta and Velocities 

10. Let />!, p 2 , etc. be the momenta, and q ly q 2J etc. 
the velocities at a given instant, and let p t , p 2 , etc., 
q x , q 2 , etc. be another system of momenta and velocities, 

such that . , ^ 

P!=np lt q 1 =nq lJ etc (10). 

It is manifest that the systems p, q will be consistent 
with each other if the systems p, q are so. 

Now let n vary by 8n. The work done by the force 
F 1 is [by 7] F J 8q 1 =$ l Bp 1 =4 1 p 1 n8n (n). 

Let n increase from o to i ; then the system is brought 
from a state of rest into the state of motion (qp), and the 
whole work expended in producing this motion is 

rl 

etc.) I ndn (12). 



rl 
But ndn=\, 



and the work spent in producing the motion is equi- 
valent to the kinetic energy. Hence 

TA = \ (Mi + Pd* + etc -) (13), 

where T p - q denotes the kinetic energy expressed in 
terms of the momenta and velocities. The variables 
q lt q 2 , etc., do not enter into this expression. 

The kinetic energy is therefore half the sum of the 
products of the momenta into their corresponding 
velocities. 

When the kinetic energy is expressed in this way we 
shall denote it by the symbol T p - q . It is a function of the 
momenta and velocities only, and does not involve the 
variables themselves. 

1 1 . There is a third method of expressing the kinetic 
energy, which is generally, indeed, regarded as the 
fundamental one. By solving the equations (3) we may 
express the momenta in terms of the velocities, and 



132 MOTION OF A CONNECTED SYSTEM [CH. 

then, introducing these values in (13), we shall have an 
expression for T involving only the velocities and the 
variables. When T is expressed in this form we shall 
indicate it by the symbol T- q . This is the form in which 
the kinetic energy is expressed in the equations of 
Lagrange. 

12. It is manifest that, since T P , T-, and 7^, are 
three different expressions for the same thing, 



or T 9 +Ti-p 1 h-pdt-etc.= o ...(14). 

Hence, if all the quantities p, q y and q vary, 



The variations op are not independent of the varia- 
tions S<7 an d <><?> s that we cannot at once assert that 
the coefficient of each variation in this equation is 
zero. But we know, from equations (3), that 
PT 1 

p-fc-o.etc ....... (.6), 

so that the terms involving the variations Bp vanish 
of themselves. 

The remaining variations S<7 and 8^ are now all 
independent*, so that we find, by equating to zero the 
coefficients of 8^, etc., 



or, the components of momentum are the differential 
coefficients of T- q with respect to the corresponding 
velocities. 

* See infra, p. 159. 



ix] LAGRANGIAN EQUATIONS 133 

Again, by equating to zero the coefficients of 8q lt etc., 



or, /re differential coefficient of the kinetic energy with 
respect to any variable q 1 is equal in magnitude but opposite 
in sign when T is expressed as a function of the velocities 
instead of as a function of the momenta. 

In virtue of equation (18) we may write the equation 

fmotion(9) 



d 9Tx an 

or *i = j- "arS - a~s ...... (20), 

dt dfa 8?j 

which is the form in which the equations of motion 
were given by Lagrange. 

13. In the preceding investigation we have avoided 
the consideration of the form of the function which 
expresses the kinetic energy in terms either of the 
velocities or of the momenta. The only explicit form 
which we have assigned to it is 

T v - q = H>>i$i + *2& + etc.) ...... (21), 

in which it is expressed as half the sum of the products 
of the momenta each into its corresponding velocity. 

We may express the velocities in terms of the differ- 
ential coefficients of T v with respect to the momenta, 
as in equation (3) [ ; thus] 



This shows that T v is a homogeneous function of the 
second degree of the momenta /> 1? p 2 , etc. 

We may also express the momenta in terms of 7 1 ,, 
and we find 

...... ( 23 ) 



134 MOTION OF A CONNECTED SYSTEM [CH. 

which shows that T^ is a homogeneous function of the 
second degree with respect to the velocities q lt q 2 , etc. 
If we write 

PH for |p-j, 

an d Qn for gT~2> Qiz for " ** ,etc, 

then, since both T- q and T v are functions of the second 
degree of q and of p respectively, both the P's and the 
Q's will be functions of the variables q only, and inde- 
pendent of the velocities and the momenta. We thus 
obtain the expressions for T, 

* T h = Piiti 2 + 2P 12 9i9 2 + etc. ...(24), 
etc. ...(25). 




The momenta are expressed in terms of the velocities 
by the linear equations 



etc ....... (26), 

and the velocities are expressed in terms of the momenta 
by the linear equations 

etc ....... (27). 



In treatises on the dynamics of a rigid body, the 
coefficients corresponding to P n , in which the suffixes 
are the same, are called Moments of Inertia, and those 
corresponding to P 12 , in which the suffixes are different, 
are called Products of Inertia. We may extend these 
names to the more general problem which is now 
before us, in which these quantities are not, as in the 
case of a rigid body, absolute constants, but are func- 
tions of the variables q lt q z , etc. 

In like manner we may call the coefficients of the 
form Q n Moments of Mobility, and those of the form 
) 12 , Products of Mobility. It is not often, however, 
that we shall have occasion to speak of the coefficients 
of mobility. 



ix] MOBILITY AND INERTIA 135 

14. The kinetic energy of the system is a quantity 
essentially positive or zero. Hence, whether it be ex- 
pressed in terms of the velocities, or in terms of the 
momenta, the coefficients must be such that no real 
values of the variables can make T negative. 

There are thus a set of necessary conditions which 
the values of the coefficients P must satisfy. These 
conditions are as follows: 

The quantities P n , P 22 , etc. must all be positive. 

The n i determinants formed in succession from 
the determinant 



*13> 



P P P P 

*ln> r 2n> -^Sn? *7 

by the omission of terms with suffix i , then of terms 
with either i or 2 in their suffix, and so on, must all be 
positive. 

The number of conditions for n variables is therefore 
2n i. 

The coefficients Q are subject to conditions of the 
same kind. 

15. In this outline of the fundamental principles of 
the dynamics of a connected system, we have kept out 
of view the mechanism by which the parts of the system 
are connected. We have not even written down a set 
of equations to indicate how the motion of any part of 
the system depends on the variation of the variables. 
We have confined our attention to the variables, 
their velocities and momenta, and the forces which act 
on the pieces representing the variables. Our only 
assumptions are, that the connexions of the system are 
such that the time is not explicitly contained in the 
equations of condition, and that the principle of the 
conservation of energy is applicable to the system. 



136 MOTION OF A CONNECTED SYSTEM [CH. ix 

Such a description of the methods of pure dynamics 
is not unnecessary, because Lagrange and most of his 
followers, to whom we are indebted for these methods, 
have in general confined themselves to a demonstration 
of them, and, in order to devote their attention to the 
symbols before them, they have endeavoured to banish 
all ideas except those of pure quantity, so as not only 
to dispense with diagrams, but even to get rid of the 
ideas of velocity, momentum, and energy, after they 
have been once for all supplanted by symbols in the 
original equations. In order to be able to refer to the 
results of this analysis in ordinary dynamical language, 
we have endeavoured to retranslate the principal 
equations of the method into language which may be 
intelligible without the use of symbols. 

As the development of the ideas and methods of 
pure mathematics has rendered it possible, by forming 
a mathematical theory of dynamics, to bring to light 
many truths which could not have been discovered 
without mathematical training*, so, if we are to form 
dynamical theories of other sciences, we must have our 
minds imbued with these dynamical truths as well as 
with mathematical methods. 

In forming the ideas and words relating to any 
science, which, like electricity, deals with forces and 
their effects, we must keep constantly in mind the ideas 
appropriate to the fundamental science of dynamics, so 
that we may, during the first development of the science, 
avoid inconsistency with what is already established, 
and also that when our views become clearer, the 
language we have adopted may be a help to us and not 
a hindrance. 

* It has also generalized our conception of dynamics, so that it 
is possible to assert that a physical system is of dynamical type 
although we may not have been able to form an idea of the con- 
figurations and motions that are represented by the variables, 
See Appendix II. 



APPENDIX I (1920) 

The Relativity of the Forces of Nature 

THE idea of the forces of nature was introduced into 
science in definite form by Sir Isaac Newton, in the 
expression of his Laws of Motion in the Introduction 
to the Principia. He specified physical force as recog- 
nized and measured by the rate at which the velocity 
of the body on which it acts is changing with the time. 
This was the simplest measure conceivable; it was 
postulated tacitly that the forces so recognized corre- 
spond to actual invariant causes of motion, which are 
always present, in accordance with the uniformity of 
nature, whenever the same conditions of the surrounding 
system of bodies recur. An underlying question is thus 
suggested as to why this particular measure corresponds 
to objective nature, and not some more complex one, 
involving for example the velocity also, or the rate of 
change of the acceleration as well as that of the velocity. 
But this introduction of the idea of forces of nature 
also gave rise to the practical need of specifying some 
definite mode of prescribing velocity and its rate of 
change. Position and velocity belong to one system of 
bodies in space and time, but are relative to some other 
system. The simplest plan is to postulate some standard 
system for general reference. Accordingly Newton laid 
down a scheme of absolute space and absolute time, 
with respect to which the movements and forces in 
nature are to be determined. It is then necessary for 
dynamical science to determine this scheme of reference 
provisionally, for the set of problems in hand, and 
continually to correct its specification as the advance of 
knowledge requires. Thus for ordinary purposes the 
space referred to the surrounding landscape and the 
time of an ordinary vibrator will suffice for a standard ; 



138 APPENDIX I 

but in wider problems when the rotation of the Earth 
has to be recognized these are no longer adequate, and 
must be replaced by a scheme of space and time which 
does not revolve with the Earth; and so on. The 
revolution effected by Copernicus, in transferring the 
centre of reference from the Earth to the Sun, was thus 
a preliminary to this dynamical order of ideas. We can 
conceive an ultimate system of space and time as that 
frame to which the stars and stellar universes can be 
related, so as to secure the greatest simplicity in the 
mode of describing their motions. Any frame of space 
and time to which the forces of nature are thus con- 
sistently referred, with sufficient precision for the 
purposes in view, has been named a frame of inertia, 
because with respect to it these forces are determined 
by the Newtonian product inertia-acceleration. For 
ordinary purposes there are many equally approximate 
frames of inertia; any uniform motion of translation 
of such a frame will make no difference in its practical 
efficacy. 

This postulation of a standard space and a standard 
time in the Principia in 1687 was made with a view to 
simple treatment of the motions of the planetary bodies 
in space: but it at once excited the criticism of philo- 
sophers both at home and abroad, though apparently 
they had no practical alternative to offer. The illustrious 
Leibniz continued to challenge its validity ; his epistolary 
controversy with Dr Samuel Clarke, who assumed on 
abstract principles the championship of the Newtonian 
practical formulation, is one of the classics of meta- 
physical philosophy. Our own Berkeley as a student at 
Trinity College, Dublin, where he was already thinking 
out his critical idealist scheme of philosophy, came up 
against the same kind of difficulties in his study of the 
foundations of the Newtonian system of the world. 
Have we any warrant for assigning an absolute frame of 
space and time for the laws of nature, especially with 
respect to the vast vacant spaces of astronomy? and 



ABSOLUTE SPACE AND TIME 139 

could we have valid means of recognizing any such 
frame ? It is perhaps largely a question of expression ; 
if philosophers could come to mean the same thing by 
the terms they use they ought to agree, otherwise the 
universal validity of the operations of the mind might 
come into doubt. 

The validity of such practical specification of a 
standard space and time has remained abstractly an 
open question; in recent years it has again come 
prominently into discussion. The phenomena of elec- 
tricity and light had been thoroughly explained, under 
the guidance of Faraday and Clerk Maxwell, in terms 
of activities established and propagated in an aether of 
space, which is at rest in undisturbed regions so that it 
is natural to fit into it the Newtonian frame of space 
and time. The aether would thus be space and time 
endowed with physical properties, inertia and elasticity, 
as well as properties of extension. But it was found 
later that very refined and delicate experiments that 
seemed qualified to determine the motion of the Earth 
relative to the aether and it must be at least of the 
order of its orbital velocity round the Sun all failed to 
show any result. This was not unexpected, and was in 
fact quite explicable on the lines of Maxwell's theory. 
But it has stimulated independent trains of thought 
which in the end have propounded the question whether 
it is possible, at the cost of more complex and pro- 
visional modes of reference, to get rid altogether of the 
universal forces of nature such as gravitation, whose 
sole evidence is the acceleration of motions for which 
they are introduced as the cause. Thus if the scale of 
time is made to alter from place to place, so that dura- 
tion is a function of position, the apparent values of 
gravitational accelerations will of course all be changed. 
The argument then is that (cf. 103) all bodies in the 
same locality possess exactly the same acceleration on 
account of gravitation : if this universal feature can be 
absorbed into a complex reckoning of space and time, 



140 APPENDIX I 

and so got rid of, the other relations of physical nature 
will merely have to become relative to the slightly 
altered reckoning introduced for this purpose. But our 
knowledge of physical extension and duration comes 
mainly from the sense of sight: little of it would have 
been acquired by a race without vision. It is impossible 
to ignore the rays of light as messengers of direction 
and duration from all parts of the visible universe. 
These essential and determining phenomena of radia- 
tion also must become mere local features of time and 
space, or else they would put us in connexion with a 
universal frame with respect to which they are propa- 
gated. However that may be, a theory which claims 
to be founded on metaphysical principles has recently 
been developed by Einstein and a numerous and 
important school, in which it is found that the forces 
of gravitation, and no other, can be represented with 
precision as inherent in a more complicated scheme of 
space and time instead of in the physical nature that 
that frame helps to describe ; while at the same time they 
thereby fall into line with the electrodynamic doctrines 
of relativity above-mentioned. 

It has been recognized however also that the same 
results can arise naturally, and without involving 
revolutionary ideas of time and space, as a slight 
(though analytically complex) expansion of the funda- 
mental physical formulation of Least Action (infra) ; the 
special relations of stress, energy, and momentum on 
which as criteria the theory had to develop being in 
fact already implicit on that universal principle. 

This alteration in the mode of expression of New- 
tonian gravitation of course makes very little practical 
difference; it however claimed special notice as re- 
moving one outstanding slight discrepancy with obser- 
vation, in the motion of the inner planet Mercury, 
which had previously to be ascribed to an assumed 
distribution of mass between the planet and the Sun. 
Such an equivalent warping of the frame of space and 



GRAVITATION AND LIGHT 141 

time must also affect either in reality or in appearance 
the propagation of radiation wherever gravitation is 
intense. One such inference is that rays of light would 
be very slightly deviated in passing close to the Sun : and 
the results of the Greenwich and Cambridge astronomers 
who observed the solar eclipse of 1919 have in fact 
confirmed the required amount rather closely. But 
another result of such an order of ideas, of a spectro- 
scopic character, still lacks any definite confirmation. 
The primary desideratum as regards gravitation was 
to find a mathematical mode of expression which would 
bring it into touch with the theory of electrical agencies 
and of radiation, from which it had been isolated, and 
even, as regards the nature of the relation of inertia to 
weight, in very slight discord. This has been done by 
ascribing the acceleration common to all bodies merely 
to an altering frame of reference, instead of the intro- 
duction into ordinary space of an intrinsic gravitational 
potential function indicating an independent type of 
local activity. For velocities very large, thousands of 
times greater than the actual speeds of the heavenly 
bodies, the results of this view would be quite different 
from the simple Newtonian gravitation, and with our 
means of expression they would be of extreme compli- 
cation : but in the actual stellar world the difference is 
excessively slight, and in the right direction. So far 
from replacing Newtonian astronomy it can only 
establish connexion with reality by making use of its 
representations and methods. We may perhaps con- 
clude that the linking up of gravitation, previously 
isolated, with other physical agencies has been effected : 
but we ought not to exclude a hope that the mode of 
expression of this connexion may in time be greatly 
simplified, especially by more attention to the Principle 
of Action, as it is only very small changes that are in- 
volved. Meantime the extrapolation, based on the pre- 
sent general formulation of the theory, to exploration 
of universes involving far higher speeds than the stars 



142 APPENDIX I 

possess in our own, is a fascinating subject for abstruse 
mathematical speculation. 

The general doctrine of relativity, at any rate in its 
more extreme formulations, impugns the validity of 
arguments such as those of 105-6. This question must 
relate to the meanings of the parties to a controversy. If 
we were shut off from sight of the stars there might be 
greater reason for claiming that it would be unphilo- 
sophic even to mention such a thing as an absolute 
rotation of the Earth, or any movement that could not 
be expressed as conditioned by adjacent bodies. That 
type of theory claims to settle all things by local scale 
and clock : but it also has in practice to requisition the 
use of the directions and periodic times of rays of light 
as valid means of discrimination. Unless the rays are 
to bend to the control of scale and clock, these measures 
will not be concordant: if they do, the connexion may 
be held to fix the frame with respect to which the rays 
travel with their assumed universal velocity, and thus 
to determine in part what has been regarded as the 
aether of space. An artificial gravitational field could 
be simulated by accelerating the frame of reference, 
provided it is not done by a mere algebraic change of 
coordinates : but the rays of light might have different 
speeds in it forward and backward, which would seem 
to involve a discriminating criterion for any such un- 
restricted "principle of equivalence " of a gravitational 
field to a changing frame. Any purely algebraic theory 
is an abstraction from the wider field of phenomena, 
and an essential question for it is the range of its own 
validity. 

Note to 145 

As the mean result of numerous modern determina- 
tions Cavendish's value 5-45 for the mean density of 
the Earth has to be increased by less than two per cent. 
The torsion apparatus has been very greatly reduced 



GRAVITATION PROPORTIONAL TO MASS 143 

in size and improved by C. V. Boys (1894) by use of 
his extremely fine and perfectly elastic quartz fibres 
for the torsional suspension. 

The Michell- Cavendish- Coulomb torsion balance 
has been applied by Eotvos to test the proportionality 
of gravitation to mass, with results of extreme precision. 
The apparent weight of a body is its gravitation to the 
Earth as modified by a centrifugal force which is 
oblique to the vertical, being directed away from the 
axis of the diurnal rotation. The latter part is of course 
considerable, being a fraction of one per cent, of the 
whole; and it has a horizontal component along the 
meridian. If the mass factors in the two parts were not 
exactly equal a torsion balance, with the ends of its 
horizontal bar loaded by masses of different substances, 
would indicate a deflection of the bar relative to its frame 
when it is turned round the vertical from east-west 
to west-east. Eotvos (1891, 1897) thus found that any 
defect of proportionality of weight to mass must actually 
be less than one part in twenty million : and Zeeman, by 
a reduced apparatus with quartz-fibre suspension, has 
recently (1917) pushed the result still lower and extended 
it to crystals and to substances of radioactive origin. 
As it happens, this is nearly to the same order as the 
optical and electric verifications of absence of effects of 
convection through the aether owing to the Earth's 
motion. 

If m is the inertia-mass of the centrifugal force and 
m' the mass which gravitates, then if m were equal to 
m' the apparent weight would be in the same direction 
for all substances and the experiment would show no 
result. Any possible result would thus be readily com- 
puted as that due to the centrifugal force of the excess 
m m',the moment of its horizontal component round 
the axis of torsion operating different ways in the two 
positions of the bar and frame. 

It is a consequence of Maxwell's electrodynamics 
that when a body loses energy e by radiation it loses 



i 4 4 APPENDIX I 

inertia of amount c/c 2 , where c is the velocity of light. 
In modern extensions of that theory all energy has 
inertia. The inertia of an electron seems to be all 
associated with its steady kinetic energy of motion. 
The closeness of the Eotvos result thus carries the 
conclusion that the inertia of an electron must all 
gravitate, and in fact that all energy possesses inertia 
which is also gravitative. Thus neither inertia nor 
gravitation could continue to be specific constants of 
matter: they must be connected up either with the 
aether in which matter subsists, or with the abstract 
reference-frame of space-time which is all that can 
remain if such a medium is denied. 



APPENDIX II (1920) 

The Principle of Least Action 

THE great desideratum for any science is its reduction to 
the smallest number of dominating principles. This 
has been effected for dynamical science mainly by Sir 
William Rowan Hamilton, of Dublin (1834-5), building 
on the analytical foundations provided by Lagrange in 
the formulation of Least Action in terms of the methods 
of his Calculus of Variations (1758), and later (1788) 
but less fundamentally for physical purposes on the 
principle of virtual work in the Mecanique Analytique. 
The principle of the Conservation of Energy, inas- 
much as it can provide only one equation, cannot 
determine by itself alone the orbit of a single body, 
much less the course of a more complex system (thus 
107-112 above need some qualification). But if the 
body starts on its path from a given position in the 
field of force and with assigned velocity, the principle 
of energy then determines the velocity this body must 
have when it arrives at any other position, either in the 
course of free motion or under guidance by constraints 
such as are frictionless and so consume no energy. If 
W, a function of position, represents the potential 
energy of a body in the field, per unit mass, the velocity 
v of the body is in fact determined by the equation 

\rmf- + mW= \mv + mW Q = mE, 
where the subscripts in v and W refer to the initial 
position; and mE is the total energy of the body in 
relation to the field of force, which is conserved through- 
out its path. Thus 

V= (2E-2W)^\ 

so that the velocity v depends, through W, on position 
alone. 

Now we can propound the following problem. By 



146 APPENDIX II 

what path must the body, of mass m, be guided under 
frictionless constraint from an initial position A to a 
final position B in space, with given conserved total 
energy mE, so that the Action in the path, defined as 

the limit of the sum ZmvSs, that is as | mvds, where Ss 

is an element of length of this path, shall, over each 
stage, be least possible? The method of treating the 
simpler problems of this kind is known to have been 
familiar to Newton : in the case of the present question, 
first vaguely proposed by Maupertuis* when President 
of the Berlin Academy under Frederic the Great, the 
solution was gradually evolved and enlarged by the 
famous Swiss mathematical family of Bernoulli and 
their compatriot Euler : and finally, extended to more 
complex cases, it gave rise, after Euler's treatise of date 
1744, in the hands of the youthful Lagrange (Turin 
Memoirs, 1758) to the Calculus of Variations, the most 
fruitful expansion of the processes of the infinitesimal 
calculus, for purposes of physical science, since the time 
of Newton and Leibniz. 

Let us draw in the given field of force a series of 
closely consecutive surfaces of constant velocity, and 
therefore of constant potential energy mW: and let us 
consider an orbit ABCD. . . intersecting these surfaces at 

the points B, C, D, We shall regard, in the Newtonian 

mannerf, the velocity as constant, say v lt in the in- 
finitesimal path from B to C, and constant, say v 2 , from 
C to D : these elements of the path are thus to be re- 
garded as straight, the field of force being supposed to 
operate by a succession of very slight impulses at B, 
C, D, ... such as in the limit, as the elements of the path 
diminish indefinitely, will converge to the continuous 
operation of a finite force. 

* The notion of an Action possibly with minimal quality, not 
merely passive inertia, as concerned in the transmission of 
Potentia or energy, is ascribed to Leibniz by Helmholtz in 1887. 

f Cf. Principia, Book i, Sec. n, Prop, i, on equable description 
of areas in a central orbit. 




LEAST ACTION 147 

If ZvSs is to be a minimum over this section ABCD. . . 
of the path, then by the usual criterion any slight altera- 
tion, by frictionless constraint, which would compel the 
body to take locally an 
adjacent course BC'D, \ A 

ought not to alter the 
value of the Action so far 
as regards the first order 
of small quantities. Now, 
on our representation of 
the force as a rapid suc- 
cession of small impulses, 
the change so produced in 
the value of this function of Action is equal to 

^ (EC' - BC) + v 2 (C'D - CD) 

hence this must vanish, up to the first order. But 
BC' - BC is equal to - CC' cos BCC', and C'D - CD 
is equal to CC' cos DCC'. Thus the condition for a 
stationary value is that the component of v^ along CC' 
is equal to the component of v 2 along the same direction, 
where CC' is any element of length on the surface of 
constant v, that is of constant W, drawn through C. 
This involves that the impulse which must be imparted 
to the body at C in order to change its velocity from 
v to v 2 must be wholly transverse to this surface: or, 
on passing to the limit, that the force acting on the 
body must everywhere be in the direction of the 
gradient of the potential W. That is, whatever the form 
of this potential function may be, the succession of 
impulses must be in the direction of its force; it is 
already prescribed by the form of v that they are of 
the amounts necessary to make changes in the velocity 
that are in accord with conservation of energy. These 
are just the criteria for a free orbit. Hence for any short 
arc of any free orbit the Action mZvSs is smaller than 
it could be if the orbit were slightly altered locally 
owing to any frictionless constraint. The free orbit is 



148 APPENDIX II 

thus describable as the path of advance that would be 
determined by minimum expenditure of Action in each 
stage, as the body proceeds: though this does not imply 
that the total expenditure of Action from one end to 
the other of a longer path is necessarily or always the 
least possible. This formula of stationary (or say mini- 
mal) Action, expressed by the variational equation 

8 mvds = o, where \mi?- + mW = mE, 

is by itself competent to select the actual free orbit 
from among all possible constrained paths. 

And generally, for any dynamical system having kinetic 
energy expressed by a function T of a sufficient number 
of geometrical coordinates, and potential energy ex- 
pressed by W, it can be shown that the course of 
motion from one given configuration to another is com- 
pletely determined by the single variational equation 

SJTdt = o subject to T + W = E, 

E being the total energy, which is prescribed as con- 
served, so that the variations contemplated in the 
motion must be due only to frictionless constraints. 
Another form of the principle is that 

7 - W)dt=o 

provided the total time of motion from the given initial 
to the given final configuration is kept constant. This 
form is more convenient for analytical purposes because 
the mode of variation is not restricted to frictionless 
constraint ; as conservation of the energy is not imposed, 
extraneous forces, which can be included in a modifica- 
tion of W, may be in operation imparting energy to the 
system. Constancy of the time of transit, which here 
takes the place of conservation of the energy, is analyti- 
cally, though not physically, a simpler form of restriction. 
From this form the complete set of general equations 



LEAST ACTION 149 

of motion developed by Lagrange (see p. 133) is 
immediately derived by effecting the process of varia- 
tion. 

If T is a homogeneous quadratic function of the 
generalized components of velocity, T^dt is a quadratic 
function of infinitesimal elements of the coordinates: 
therefore the first form when expressed (after Jacobi) as 



does not any longer involve the time. It thus determines 
the geometrical relations of the path of the system 
without reference to time ; for a simple orbit it reduces 
to the earliest form investigated above. 

In the modern discussions of the fundamental prin- 
ciples of dynamics, especially as regards their tentative 
adaptation to new regions of physical phenomena 
whose dynamical connexions are concealed, this prin- 
ciple of variation of the Action, which condenses the 
whole subject into a single formula independent of any 
particular system of coordinates, naturally occupies the 
most prominent place. 



As a supplement to Chapter IX, these statements of 
the Principle of Action will now be established for a 
general dynamical system. This can be done most 
simply and powerfully by introducing the analytical 
method of Variations, invented by Lagrange as above 
mentioned. 

The principle, as already deduced for the simplest 
case, relates to the forms of paths or orbits : if it is also 
to involve the manner in which the orbits are described 
the time must come in. The criterion of a free path was 

that 8 I vds = o with energy E constant throughout the 
motion : it is the same as 8 tfdt = o under the same 



150 APPENDIX II 

condition; or, writing T for the kinetic energy 

it is 8 I zTdt = o under the same restriction to constancy 

of the total energy. 

Let us conduct the variation directly from this latter 
form, but now keeping the time unvaried, 



f (dx dSx dy dSy , dz dSz\ , t 
= ] m (dt^ + dt^t + dtW) dt 



in which d is the differential of x as the body moves 
along its orbit with changing time, but Sx is the 
variation of the value of x as we pass from a point on 
the orbit to a corresponding point on the adjacent pos- 
sible path that is compared with it. The introduction of 
different symbols d and 8 to discriminate these two 
types of change is the essential feature of the Calculus 
of Variations : we have already used the fundamental 
relation 8dx = dSx. Integrating now by parts, in order 
to get rid of variations of velocities which are not inde- 
pendent variations and so not arbitrary, we obtain 



[= 



dx <j, dy <j. dz 

JT ox -f- m -j~ oy -\- tn -j- 
dt dt * dt 



in this the first term represents the difference of the 
values at the upper and lower limits of the integral, 
indicated by subscripts 2 and i, which correspond to 
the final and initial positions of the body. The second 
term is equal to 



-(X8x+ 
where (X, Y, Z) is the effective force acting on the par- 



LEAST ACTION 151 

tide w, as determined by the acceleration which the 
particle acquires. 

We can extend this equation at once to any system of 
particles in motion under both extraneous and mutual 
forces. If there are no forces exerted from outside the 
system, but only an internal potential energy expressed 
by a function W, then the work of the internal forces 
of the system tends to exhaust this energy, so that 

2 (XSx + YSy + XSz) = - SW, 

and this holds good whether the algebraic equations 
expressing the constraints contain t or not. 

Thus if T now represents the total kinetic energy, 
and all the forces are internal, we can write, for variation 
from a free path to any adjacent path by frictionless 
constraint, and with times unvaried, 



Strictly, this result has been obtained for a system of 
separate particles influencing each other by mutual 
forces. It is natural to expand it to any material system 
consisting of elements of mass subject to mutual forces, 
thus including the dynamics of elastic systems. The 
ultimate analysis of the element of mass is into mole- 
cules or atoms in a state of internal motion: that 
final extension would include the dynamical theory of 
heat. 

We can now express all the coordinates x, y, z of 
the particles or elements of mass in terms of any suffi- 
cient number of independent quantities 6, (/>, ^, ... 
which determine the position and configuration of the 
system as restricted by its structure. Their number 
is that of the degrees of freedom of the system. The 
equations which express x, y, z in terms of them may 
involve / explicitly, for the equation of virtual work 
involves the displacements possible at given time ; thus 
the new form of T W can contain t . Then we can 



152 APPENDIX II 

assert that when / is not varied, and the time limits 
tj_ and t z are therefore constant, 



when the frictionless variation is taken between fixed 
initial and final positions of the dynamical system. 

This quantity T W is the Lagrangian function L 
defining by itself alone the dynamical character of the 
system : the function Lor W T is thus the potential 
energy W as modified for kinetic applications, and has 
been appropriately named by Helmholtz the kinetic 
potential of the system. Thus the particular case of 
a system at rest is included: for 



or ^Wdt is equal to BWJdt 



as W remains constant during the time: hence the 
equation of Action asserts in this case that 

SfP-o, 

which comprehends the laws of Statics in the form 
that the equilibrium is determined by making the 
potential energy stationary. For stability it must be 
minimum. 

Again, as Lis expressed as a function of the generalized 
coordinates 6, <f>, ... and their velocities, 



where # represents -j- , and 80 is equal to j 80 ' thus 

integrating by parts as before 

2T 2 



As the left side vanishes, when the terminal positions 



LEAST ACTION 153 

are unvaried, for all values of the current variations 
80, 8c/>, ..., and these are all independent and arbitrary, 
the coordinate quantities 6, </>, ... being just sufficient 
to determine the system, the coefficient of each of these 
variations must vanish separately in the integrand. Thus 
we obtain a set of equations of type 
d_dL _3L = 
dt 36 W ~ 

which are the Lagrangian equations of motion of any 
general dynamical system (20, p. 133 supra). If there are 
in addition extraneous forces in action on the system, 
the appropriate component force F e , defined as that 
part whose work F 6 86 is confined to change of the one 
coordinate 6, must be added on the right-hand side. 
These applied forces may vary with t in any manner: 
they can be merged in W by addition of terms F 6 B ... 
to it: their presence will prevent the energy of the 
system from remaining constant. 

If we restrict this comparison of paths to variation from 
a free path of the system to adjacent free paths, we have 



i f Ldt = 
It, 



now as an exact equation, and so capable of further 
differentiation ; and it provides the basis of the Hamil- 
tonian theory of varying Action. 

It will be convenient at this stage to remove the 
restriction that the time is not to be varied : to allow for 
this change we must substitute in the equation in 
place of 86 the expression 86 68t which deducts from 
the total variation of 6 that part of it which arises from 
the motion in the interval of varied time 8t. We must 
also add L8t in order to include in the time of transit 
the new interval of time 8t added on at the end by 
the variation. Thus now 



8 f L8t = L8t + *.- (86 - 68t) 



154 APPENDIX II 

Also L = T W\ and T being a homogeneous 
quadratic function, 



( 

hence 8 f Ldt = d -k 80 + . 8<f> + ... - ESt 
Jit 38 dj> 

where E is the final value of the total energy T + W. 

When no extraneous forces are supposed to be in 
action E is constant at all times : thus 



Hence, transposing the last term, the alternative form 
arises, 



ft 

I 



for variations throughout which the energy is conserved. 
This is the generalization of the previous form 

8 mvds = o for a particle, except that now the time also 

is involved, and is determined as dAJdE, where A is the 
time integral of 2 T as expressed in terms of initial and 
final configurations and the conserved energy. 

This involves the analytical result that if 0, O, ... are 
the momenta* corresponding to the coordinates 6, <, ..., 
then there must exist a certain function A (of form 
however that is usually difficult to calculate) of 
B y <f>, ... E, such that in varying from the free path to 
adjacent free paths of the system, 



A more explicit and wider form, especially for optical 
applications, is immediately involved in this formula, 

* The subscript notation of Chapter ix would here be incon- 
venient. 



LEAST ACTION 155 

that there is a function A |J of the initial and final con- 
figurations of the system and the energy, such that 



... + (t 2 - *j) SE. 

There also exists a function Pl^ of the final and initial 

r<2 
coordinates and the time , equal in value to | (T-W)dt, 



such that 



- 0^ - O^ - ... - E 2 St 2 + 

on varying from any free orbit to adjacent free orbits ; 
but now as there is no restriction to E remaining con- 
stant along an orbit, the forces may be in part extraneous 
forces whose work will impart new energy to the system. 
The mere fact that such a function P or A exists 
involves a crowd of reciprocal differential relations 
connecting directly the initial and final configurations 
of the system or a group of systems, of type such as 



which are often the expression of important physical 
results. Moreover in the form of SP, and therefore in 
such resulting relations, the final set of coordinates may 
be different from the initial set. 

The influence of disturbing agencies on any dy- 
namical system, whose undisturbed path was known, is 
by these principles reduced to determining by approxi- 
mation (from a differential equation which it satisfies) 
the slight change they produce in this single function 
P or A which expresses the system, a method perfect 
in idea but amenable to further simplifications in 
practice. 

This beautiful theory of variation of the Action from 
any free path to the adjacent ones was fully elaborated 
by Hamilton in a single memoir in two parts (Phil 



156 APPENDIX II 

Trans., 1834 and 1835), and soon further expanded in 
analytical directions by Jacobi and other investigators. 
It brings a set of final positions of a dynamical system 
into direct relations with the corresponding initial posi- 
tions, independently of any knowledge whatever of the 
details of the paths of transition. In connexion with the 
simplest case of orbits it has been characterized by 
Thomson and Tait as a theory of aim, connecting up, 
so to say, the deviations on a final target, arising from 
changes of aim at a firing point, with the correspond- 
ing quantities of the reversed motion. In geometrical 
optics, from which the original clue to the theory came, 
where the rays might be regarded as orbits of imagined 
Newtonian corpuscles of light, it involves the general 
relations of image to object that must hold for all types 
of instrument, as originally discovered by Huygens and 
by Cotes. Its scope now extends all through physical 
science. 

In certain cases the number of coordinate variables 
required for the discussion of a dynamical problem can 
be diminished. Thus if the kinetic potential involves 
one or more coordinates only through their velocities, 
the corresponding equations of motion merely express 
the constancy throughout time of the momentum that 
is associated with each such coordinate: this holds for 
instance for the case of freely spinning flywheels 
attached to any system of machinery, and for all other 
cases in which configuration is not affected by the 
changing value of the coordinate. In all such cases the 
velocity can be eliminated, being replaced by its 
momentum which is a physical constant of the motion. 
The kinetic potential can thus be modified (Routh, 
Kelvin, Helmholtz) so as to involve one or more variables 
the less, but still to maintain the stationary property of 
its time-integral. It is now no longer a homogeneous 
quadratic, but involves terms containing the other 
velocities to the first degree, multiplied of course by 
these constant momenta as all the terms must be of the 



LEAST ACTION 157 

same dimensions. Every such kinetic potential belongs 
to a system possessing one or more latent unchanging 
(steady) motions ; and a general theory of this important 
physical class of systems, and of the transformation of 
their energies, arises. 

Infactif L' = L-T^-... 

where ^ ... are a group of coordinates and ^Y ... the 
related momenta, then 



in which the first term vanishes identically, while 8 X L 
is the variation of L with regard to the remaining 
variables. Hence if L do not involve the coordinates 
/r . . . , so that T ... are constant and are not made subject 
to variation, and $ ... are eliminated from L' by intro- 
duction of T, ... then 

3 \L'dt = | 080+ 080 + ... - ESt 

depending only on the variations of the explicit co- 
ordinates at the limits, provided *F ... are kept un- 
varied, or the flywheels of the system are not tampered 
with. 

Although the cyclic coordinates do not appear at all 
in L, yet it is only in terms of L' modified as here that 
we can avoid their asserting themselves in the domain 
of varying Action. 

The ultimate aim of theoretical physical science is to 
reduce the laws of change in the physical world as far 
as possible to dynamical principles. It is not necessary 
to insist on the fundamental position which the kinetic 
potential and the stationary property of its time-integral 
assume in this connexion. Two dynamical systems 
whose kinetic potentials have the same algebraic form 



158 APPENDIX II 

are thoroughly correlative as regards their phenomena, 
however different they may be in actuality. If any range 
of physical phenomena can be brought under such a 
stationary variational form, its dynamical nature is 
suggested: there still remains the problem to extricate 
the coordinates and velocities and momenta, and to 
render their relations familiar by comparison with 
analogous systems that are more amenable to inspection 
and so better known. 



Note on Chapter IX, 9. 

It has appeared above, as Lagrange long ago em- 
phasized, that the principle of Conservation of Energy 
can provide only one of the equations that are required 
to determine the motion of a dynamical system. It 
follows that the reasoning of this section ( 9), which 
seems to deduce them all, must be insufficient. The 
argument there begins by supposing the system to move 
in any arbitrary way; that is, it assumes motions deter- 
mined by the various possible types of frictionless 
constraint that are consistent with the constitution of 
the system. The equation (9) is then derived correctly 
from (7) and (8), as the variations Bq are fully arbitrary. 
But the imposed constraints introduce new and un- 
known constraining forces which must be included in 
the applied forces F r \ and they would make the result, 
so far as there demonstrated, nugatory. 

The equations (9) are however valid, though this 
deduction of them fails. As explained above, the La- 
grangian equations (20) are derivable immediately from 
the Principle of Least Action, independently established 
as here: and then the equations (9) can be derived 
by reversing the argument. 



LEAST ACTION 159 

The procedure of 12 seems to lead to a noteworthy 
result. It asserts that if 

F T 4- T- -2.T 

* * j> T- -L q *J- vq 

then the single relation 



involves all the equations connecting coordinates, 
velocities and momenta in the system. This will remain 
true when the three sets of variables, regarded still as 
independent, are changed to new ones by any equations 
of transformation, so that this threefold classification 
into types becomes lost. Now there are cases in which 
the steady motion of a system, or an instantaneous 
phase of a varying mode of change, can be thoroughly 
explored experimentally, leading to the recognition say 
of yi physical quantities of which only 2 can be 
independent; but it is not indicated by our knowledge 
how we are to deduce from them a scheme of n coordi- 
nates, n corresponding velocities, and n momenta. We 
have arrived at the result that in every such case a 
function F must exist, and is capable of construction, 
such that 8F = o provides a set of %n equations con- 
taining all the knowledge that is needed. The relations 
(treated after Maxwell) of a network of mutually 
influencing electric coils carrying currents would form 
an example. 

In cognate manner we may assert another type of 
equation of Variation of Action 



where T v ' q = |2gp, containing n coordinates q, their 
n velocities q and their n momenta/). For this equation 
is equivalent to 



160 APPENDIX II 

leading on integration by parts as usual to two sets of 
relations of the types 



dp 



4 = _??_ 

dt~ ~d 



if in it the momenta and coordinates are regarded as 
independent variables. As ->p = ^p by (18), the 

second set are the Lagrangian dynamical equations (20). 
Thus we have here a single function 

involving coordinates and their velocities, linear in the 
latter, and an equal number of quantities/) of the nature 
of momenta, the coordinates and momenta being thus 
independent variables, such that the relation 

# = o 

leads both to the identification of the relations in which 
the momenta stand to the coordinates and to the 
dynamical equations of motion of the system. 

This result is virtually the same as equation 12 a in 
Hamilton, Phil. Trans., 1835, p. 247. In Helmholtz's 
memoir on Least Action, Crelle's Journal, vol. 100 
(1886), Collected Papers, vol. iii, p. 218 another function 
is introduced, apparently with less fitness, in which the 
velocities are regarded as independent of their coordi- 
nates but the momenta are the gradients of L with 
regard to the velocities. Cf. also Proc. Lond. Math. Soc., 
1884. 

A main source of the great power of these dynamical 
relations of minimal or stationary value, as exploring 
agents in physical science, is that the results remain 
valid however the physical character of the functions 
involved may be disguised by transformation to new 
variables, given in terms of the more fundamental 
dynamical ones by any equations whatever. This func- 



LEAST ACTION 161 

tion (j> may thus be expressed in terms of 2 quantities 
which are in any way mixed functions of coordinates 
and momenta and their gradients with respect to time 
remaining a linear function of the latter and subject to 

other limitation and the equation 8 <f>dt = o will still 

subsist and will express all the dynamical relations of 
the physical system. 

The existence of a variational relation of this type 
may be taken as the ultimate criterion that a partially 
explored physical system conforms to the general laws of 
dynamics ; while from its nature the coordinate quantities, 
in terms of which the configuration and motion of the 
system happen to be expressed, shrink to subsidiary 
importance. 



INDEX 



The numbers refer to pages 



Absolute space and time, 139 

Acceleration, 23 

Action, 145 

Attraction, 41, 66; mutual, 44 

Berkeley, G., 138 
Boys, C. Vernon, 143 

Calculation, 119 

Cavendish, H., torsion balance, 

116, 143 
Centre of mass, 45; motion of, 

47; motion referred to, 106 
Centrifugal force, 93 
Circular motion, 92 
Configuration, 3 
Conservative system, 54 
Coordinates, 124 

d'Alembert, J. le R., 49 
Descartes, R., 9 
Displacement, 15 
Dynamical system, 123; test 
of, 136, 159 

Elasticity, 83 

Elliptic orbits of planets, 108 
Energy, 158; conservation of, 
55 ; history, 75 ; not identifi- 
able, 89; latent, 90; poten- 
tial, 58, 65, 67, 77, 113; 
change to kinetic, 59; speci- 
fication of, 71; formulae for 
kinetic, 131 

Faraday, M., 139 

Force, 5, 35, 80; and mass, 
32; applied and effective, 
49; derived from energy, 
68, 88; measured by vibra- 
tions, 95 

Forces act independently, 37, 

39, 7 
Frame, Newtonian, 9, n, 138 



Gravitation, universal, 82; law 
of, 109, in; of the Moon, 
115; cause of, 121; absorbed 
into time, 142; influence on 
light, 141; proportional to 
inertia, 34, 143 

Gravity, measurement of, 101 

Hamilton, W. R., 155; his 

dynamical equations, 129 
Heat as energy, 73 
Hodograph, 107 

Impulse, 37; work of, 127 
Inertia, 29; frame of, 30; 
specification of, 134; in- 
creases with speed, 141; of 
energy, 144 

Kepler's laws, 106 

Kinetic energy, calculation of, 

62; limits of available, 63; 

of planets, 113 
Kinetic potential, 152 



Lagrange, J. L., 146; 
dynamical equations, 



his 
133. 
153 

Latent motions, 156 
Laws of motion, 27, 137 
Laws of nature, 13 
Least Action, 145; for orbit, 
147; fundamental in physics, 



Lei 



157, 160 

sibniz, G. W., 138 



Mass, measure of, 33; vector, 
44; centre of, 45; and force, 
49 

Mass-area, 50, 105 

Material system, 2, 89 

Medium, physical, 67 

Mobility, specification of, 134 

Moment, 51 



INDEX 



163 



Momentum, 38, 56; vector, 46; 
change of, 48; angular, 51; 
conserved, 53; general com- 
ponents, 126, 131; in terms 
of velocities, 132 

Newton, I. passim ; his method, 
122 

Orbit, 107, 147 

Pendulum, 97; solid reversible, 
99; Foucault's, 87; conical, 

87 

Physical Science, i, 71, 74, 161 
Position, 7 
Potential, kinetic, 152 

Rates of change, 20, 24 
Reaction, 40, 70 
Reciprocal relations, 155 



Relativity, general, 22, 25, 

29, 67, 82, 83, 137 
Rotation, test of absolute, 84; 

by gyrostat, 86 

Space and time, 9, 31, 138 
Strain, 40 
Stress, 27, 40 

Units, 35 

Variations, method of, 150 
Varying action, 153 
Vectors, 4; addition of, 6 
Velocities derived from mo- 
menta, 129 

Velocity, 19; diagram, 20 
Vibrations, 94; as measure of 
force, 95; isochronous, 95; 
counting of, 102 

Work, 54, 56, 75, 127 



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